TATE  NO'  -J- 


MATHEMATICAL  CONSTEUCTION 

INFORMAL  NUMBER  WORK 
FOE  BUSY  HANBS 

GRADES  ONE  AND  TWO 


BY 

N.  LOUISE  LAFFIN 

OF    THE    CHICAGO    SCHOOLS 


A.  FLANAGAN  COMPANY 

CHICAGO 

40341 


Copyright  1911 

BY 

N.  LOUISE  LAFFIN 


L_3 


TABLE  OF  CONTENTS 

Introduction    5 

Plans  of  Lessons 12 

All  the  combinations  to  12,  and  the  lessons  in  which 

they  are  found 16 

FOLDING. 
LESSON.  PAGE. 

I     Shawl    23 

II     Booklet   25 

III  Wall-Pocket    26 

IV  Soldier's   Cap 28 

V     Fireman's  Cap 29 

VI     Envelope    30 

VII     Picture    Frame 32 

VIII     Salt  and  Pepper  Holder 35 

IX     Boat    36 

X     Square   Prism    37 

XI     Cube    40 

XII     Cradle 42 

XIII  Chair 43 

XIV  Buggy    45 

XV  to  XXV     Baskets 46 

FOLDIN'G   AXD    WEAVING. 

I     Making    2    cuts    on    mat    (16    squares), 

weaving  with  one  strip 53 


CONTENTS 
LESSON.  PAGE. 

II  Making   2    cuts   on   mat    (12    squares), 

weaving  with  one  strip 53 

III  Making  3  cuts  on  mat,  weaving  with  2 

equal   strips 54 

IV  Making  3  cuts  on  mat,  weaving  with  4 

equal   strips 54 

V     Making  5  cuts  on  mat,  weaving  with  2 

equal   strips '55 

VI     Making  5  cuts  on  mat,  weaving  with  4 

equal   strips ^5 

VII     Making  7  cuts  on  mat,  weaving  with  6 

equal  strips 56 

VIII     Making  5  cuts  (relation  1  to  2),  weaving 

with  4  strips  (relation  1  to  2) 56 

IX     Making  5  cuts  (relation  1  to  2),  weaving 

with  5  strips  (relation  1  to  2) 57 

X     Making  4  cuts  (relation  1  to  4),  weaving 

with  3  strips  (relation  1  to  4) 57 

XI     Making  7  cuts  (relation  1  to  2),  weaving 
with  7  strips  (relations  1  to  2  and  1 

to  4)  over  2  and  under  1 58 

XII     Eelation  1  to  2  cutting  mat,  1  to  4  cut- 
ting strips,  weaving  over  3  and  under  1.     58 

FREE  CUTTING  AND  WEAVING. 

I  Eelation  1  to  2 60 

II  Relation  1  to  4 61 

III  Relation  1  to  3  (4  cuts) 61 

IV  Relation  1  to  3  (6  cuts) 62 


CONTENTS 
LESSON".  PAGE. 

V  Relation  1  to  3  (8  cuts) 62 

VI     Relation  1  to  3  (7  cuts) 63 

VII     Relation  1  to  5,  1  to  2  (6  cuts) 63 

WALL-PAPER  MADE  BY  FREE-CUTTING ....       65 
PAPER  RINGS  MADE  BY  FREE-CUTTING 67 

MEASURING  AND   WEAVING. 

I     Bank  Decoration.   (Rulers:  V\  2",  3", 

4")     68 

II     Calendar  Back.  (Rulers:  1",  2",  3",  4", 

7")    '. ...     69 

III  Blotter.    (Rulers:   Q" ,  5",  4",  3",  2", 

1")     .'     70 

IV  Xapkin  Ring.  (Rulers:  I",  2",  3",  A", 

5",    6") 72 

V  Telephone  Pad.  (Rulers:  1",  4",  5",  8")     73 
VI     Wall-pocket  for  Letters.  (Rulers:  1",  2", 

3",  7",  10'') 74 

VII     Circular  Woven  Basket.  (Rulers:  1",  2", 

5",  G",  9",  10") 76 

VIII     Xeedle-book.     (Rulers:  1",  3",  V,  7". 

8",  10") .'     78 

PROGRESSIVE    MEASURING    LESSONS. 

Introduction  of 
Single  Unit  Rulers. 

I     Scissors  Holder.   (Ruler :  8") 80 

II     Mayflower.  (Rulers :  8",  4") 81 

III     Cup.  (Rulers  :  6",  3") 83 


CONTENTS 
LESSON.  PAGE. 

IV     Book-mark.  (Eulers :  4'',  2'') 85 

V     Stamp  Pocket.  (Rulers:  2",  3",  5")..,  86 
VI     Washcloth  Pocket.  (Rulers:  1",  7",  3'% 

4'')     88 

VII     Pilgrim's  Bonnet.   (Rulers:  2",  4",  6'')  89 

VIII     Stove   (Rulers:  3",  6",  9") 91 

IX     Bank.  (Rulers :  4",  8",  12") 93 

X     Envelope  (for  decoration  units)  (Rulers: 

5",  1",  7",  2") 95 

XI     Taboret.  (Rulers:  5",  3",  8'',  1") 96 

XII     Father   Bear's    Chair.    (Rulers:   3',    Q>", 

9",    12") 97 

XIII  Mother's   Chair.    (Rulers:    2",   4",    6", 

8")    99 

XIV  Baby's  Chair.  (Rulers:  V,  2",  3",  4").  100 
XV     Father's  Bed.  (Rulers:  3",  6",  9",  12").  101 

XVI      Mother's  Bed.  (Rulers:  2",  4",  6",  8").  103 

XVII      Baby's  Bed.  (Rulers:  1",  2",  3",  4") .  .  103 
XVIII     Father  Bear's  Bowl.    (Rulers:   12",  4", 

3",  1") 103 

XIX     Mother's   Bowl.    (Rulers:   12",   3",   2", 

1")     104 

XX     Baby's  Bowl.  (Rulers:  12",  2",  1")...  105 

XXI     Jack's  Pail.  (Rulers:  9",  5",  4",  1") .  .  106 

XXII     Jill's  Pail.   (Rulers:  9",  3",  2",  1")..  107 

XXIII  Fox's  Dish.  (Rulers:  5",  2",  1") 108 

XXIV  Stork's  Dish.  (Rulers:  5",  7",  6",  1").  109 
XXV     Handbag.  (Rulers:  9",  6",  3",  2",  1").  109 

XXVI     Pencil-Box  With  Lid.  (Rulers:  10",  2", 

3",  5") Ill 


CONTENTS 

LESSON.  PAGE. 

XXVII     Match-Safe.    (Rulers:   5",  4'',  2,",   2", 

1")    112 

XXVIII     Wood-Box.  (Rulers: 
3",  6",  9",  12" 

2",  4",  6",  8") 113 

XXIX     Sled.  (Rulers:  y,  4",  3",  2",  1") ....   114 

XXX     Pushcart.  (Rulers:  1'\  5",  2",  V) 115 

XXXI      Gocart.   (Rulers:  2",  4",  6",  8") 117 

XXXII      Cradle.  (Rulers:  1",  9",  5",  4'') 119 

XXXIII  Bureau.  (Rulers:      3'%  6",  9",  12" 

8'',  2",  5") 121 

XXXIV  Chiffonier.    (Rulers:    2",    4",    6",    8", 

10'')     123 

XXXV     A  Large    Envelope.    (Rulers:    11",    1", 

8",  4") 125 

XXXVI     Book    for    Cuttings — unfolded    sheets- — 

(Rulers :  1",  3",  6",  9") 127 

XXXVII     Book  for  Words— unfolded  sheets— (Rul- 
ers: 6",  V%  7") 128 


MATHEMATICAL  CONSTRUCTION 


INTRODUCTIOX 

All  the  construction  work  should  be  carefully  selected  by 
the  teacher  with  a  two-fold  purpose. 

1st.  The  appropriateness  of  the  thing  itself  to  the  child's 
need;  a  co-relation  to  other  studies,  or  a  gift,  plaything  or 
actual  necessity. 

2nd.  The  appropriateness  of  the  number  relations  to  the 
child's  mathematical  discoveries;  a  proper  sequence  from  the 
simplest  of  mathematical  relations  to  the  more  complex,  and 
a  proper  order  of  inferences. 

The  possible  number  relations  found  in  the  process  of 
the  construction  of  an  article  sliould  be  brought  out  carefully 
by  the  teacher.  Wliile  the  child  is  making  something  he  is 
in  a  receptive  mood  for  this  number  work.  Each  step  gives 
new  surfaces  or  lines  to  compare. 

When  the  teacher  artfully  questions,  giving  concrete 
problems  regarding  these  ever  changing  lines  and  surfaces, 
she  gets  answers  from  alert  minds ;  minds  that  are  comparing 
and  coming  to  conclusions  then  and  there;  minds  that  are 
not  trying  to  remember  what  the  teacher  said  yesterday  or  a 
week  ago;  minds  that  are  seeing  and  drawing  inferences  at 
tile  present  moment. 

In  applying  number  work  to  the  making  of  things  in  the 

5 


6  MATHEMATICAL  CONSTEUCTION 

first  and  second  grades  (by  folding,  free  cutting  or  measur- 
ing), a  teacher  should  let  the  child  judge  the  relations  of 
things  himself.  By  repeated  acts  of  judgment  he  gains  the 
power  to  see  relations  correctly. 

Indefinite  relations  are  seen  before  the  definite.  There- 
fore, the  earliest  lessons  (found  in  the  folding  of  articles) 
should  deal  only  with  words  which  express  contrast;  such 
as  larger  and  smaller,  longer  and  shorter,  more  or  less.  A  few 
weeks  of  this  indefinite  relation  work  usually  suffice.  Then 
follow  the  relations,  1,  2,  3,  4,  equal  ^,  i,  i-  In  these 
lessons  the  teacher  should  only  show  the  fold,  the  child 
imitating.  This  gives  mental  training  through  sight.  After 
the  child  has  imitated  the  teacher's  silent  direction,  the 
teacher  should  ask  questions  bringing  out  the  relations 
(indefinite  or  definite  as  the  state  of  the  child's  mind 
permits). 

These  questions  should  introduce  technical  words  inci- 
dentally and  informally  to  the  pupil.  By  giving  little  concrete 
problems  based  on  the  relation  of  surfaces,  lines  and  solids, 
she  can  do  this.  For  instance :  when  a  square  is  folded  into 
equal  oblongs  she  may  say,  "If  this  oblong  is  enough  cloth 
(pointing  to  one  oblong)  for  one  doll's  dress,  what  is  this 
oblong?"  (pointing  to  other).  Or,  "Play  this  oblong  is  a 
dresser  scarf,  which  has  lace  all  around  it.  If  the  lace  on 
the  upper  edge  costs  a  dime,  what  does  the  lace  on  the  side 
edge  cost?"  Ans.  "More  than  a  dime."  Child  gets  words 
oblong,  upper  edge,  lower  edge  incidentally. 

Giving  concrete  problems  in  this  way  is  interesting  to 
the  imaginative  mind  of  a  child.  He  plays  that  sand  is 
sugar,  buttons  are  money,  and  stones  are  potatoes  or  apples. 
Why  can't  he  play  that  oblongs  or  squares  are  parks,  play- 


INTRODUCTION  7 

grounds,  ceilings,  bed  quilts,  tablecloths,  articles  of  clothing, 
etc.? 

And  why  can't  he  also  play  that  two  of  those  are  two 
parks,  or  three  of  them  are  three  parks?  If  one  park  con- 
tains five  acres,  two  parks  contain  two  five  acres,  etc.  Why 
can't  he  play  that  a  triangle  is :  a  doll's  shawl,  a  pond  of  ice 
to  skate  on,  a  tile,  a  piece  of  glass,  the  leather  corner  of 
a  desk  blotter,  etc.  ?  Why  can't  he  play  that  edges  require 
fringe  for  rugs,  bedspreads,  napkins,  doilies,  the  top  of  a 
carriage,  etc.;  fences  for  parks,  farms,  yards,  empty  lots, 
etc. ;  binding  for  slates,  books,  pictures,  cloth ;  curbstones  for 
streets;  hedges  or  trees  in  a  row  for  yards,  parks,  roads? 

When  the  children  are  familiar  with  the  technical  words, 
the  teacher  should  give  verbal  directions  instead  of  silently 
showing  them,  thus  giving  child  mental  training  through 
hearing.  These  verbal  directions  are  more  difficult  than  the 
sight  directions,  for  they  involve  a  memory  of  the  technical 
words  and  number  relations  learned  incidentally  through 
the  concrete  problem  work. 

After  a  child  can  fold,  by  following  the  verbal  directions 
of  the  teacher,  he  is  ready  to  use  a  ruler.  An  ordinary 
12-inch  ruler  is  beyond  the  comprehension  of  the  beginner. 
The  inch  is  within  the  2-inch  measure,  and  within  the  3-inch 
unit  there  is  the  2-inch  and  also  the  1-inch  unit.  The  child 
cannot  see  the  relation  of  1  inch  to  2  inches  on  the  ruler; 
therefore,  he  should  use  single  unit  rulers.  A  teacher  sliould 
have  a  complete  set  for  each  child  in  the  room.  A  set  consists 
of  12  pieces  of  strong,  tough,  smooth  cardboard  (pressboard 
is  the  proper  name).  Each  piece  is  1  inch  wide,  and 
respectively,  1,  2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  and  12  inches 
long. 


MATHEMATICAL  CONSTRUCTION 


UD 


Of  course  the  child  is  not  given  the  whole  set  at  once. 
He  would  be  confused. 

If  the  teacher  intends  making  something  which  requires 
a  6-inch  square,  for  instance,  she  gives  him  a  6-inch  ruler. 
If  that  must  be  bisected  he  is  given  a  3-inch  ruler.  If  the 
original  paper  is  9''xl2"  let  him  estimate  the  width  of  it. 
He  will  tell  you  that  the  width  is 

the  sum  of  3''  and    6''  or 

the  sum  of  6''  and  i/o  of  6''  or 

three  3" 
He  will  say  that  the  length  is : 

four  3''  or 

the  sum  of  6''  and  6''  or 

the  sum  o*  6''  and  3''  and  3"  or 

the  sum  of  three  3''  and  1/2  of  6" 


INTRODUCTION  9 

With  these  rulers  one  obtains  mathematical  relations  of 
the  measures  which  one  could  never  get  from  little  children 
with  the  ordinary  ruler,  which  has  the  units  within  many 
other  units.     The  children  see : 

the  sum  of  3''  and  3", 
the  difference  between  6"  and  3", 
the  division  of  6"x2, 
the  multiplication  of  3"x2. 
"When  in  a  later  lesson  the  9-inch  ruler  is  introduced,  the 


children 

see: 

6" 

3'3"=9" 

i  of  9"=3' 

3" 

9" 

9" 

9" 

-6" 

-3" 

3''  6" 

In  making  an  article  based  on  a  square  consisting  of  16 
squares,  we  use  the  dimensions : 

3,  6,  9,  12,  or 

2,  4,  6,  8,  or 

1,  2,  3,  4. 

Why  should  a  child  be  bothered  with  all  the  units  within 
units  of  an  ordinary  ruler  when  he  can  be  given  something 
which  is  definite  to  him  ?  How  easy  it  is  to  see  that  6  is  ^ 
of  12;  3  is  :|  of  it,  etc.,  with  these  distinct  units  of  measure. 

The  making  of  an  article  based  on  a  square  consisting 
of  9  squares  vividly  shows  the  relation  of  1,  2,  3;  2,  4,  6; 
3,  6,  9;  4,  8,  12.  ' 


10  MATHEMATICAL  CONSTRUCTION 

One  based  on  a  square  consisting  of  4  squares  gives  the 
relations  1,  2 ;  2,  4 ;  3,  6 ;  4,  8 ;  5,  10 ;  6,  12. 

The  first  step  in  the  using  of  the  ruler  is  like  the  first  step 
in  folding.  The  teacher  should  show  the  direction  and  the 
child  imitate  (sight  and  mind  training).  The  only  verbal 
expression  of  the  teacher  is  the  naming  of  the  ruler,  viz. : 
*'This  is  a  6-inch  ruler." 

The  second  point  in  the  use  of  the  ruler  is  that  the  teacher 
directs  verbally,  after  the  child  has  learned  how  to  use  it. 

Third  point:  Child  must  do  mental  addition,  subtrac- 
tion, division,  or  multiplication,  by  following  teacher's  verbal 
direction. 

Teacher  asks  questions  like  these : 

(Suppose  child  to  have  rulers  3",  6",  9",  12"  long,  and 
that  he  is  familiar  with  their  relations  through  former  work 
in  measuring,  and  that  he  has  paper  9"xl2".) 

Show  me  the  edge  that  is  three  3"  long. 

How  many  inches  long  is  it? 

Show  me  the  edge  that  is  the  sum  of  9"  and  3"  long. 

How  many  inches  long  is  it? 

On  the  upper  long  edge  mark  off  ^  of  12''.  One-half  of 
12"  is  how  many  inches? 

If  it  costs  $12.00  to  build  a  fence  on  this  edge  (pointing 
to  the  12-inch  edge)  what  part  of  $12.00  will  it  cost  to  build 
a  fence  on  this  part  (showing  6")  of  it?    Ans..    ^  of  $12.00. 

How  much  money  will  it  cost?     ($6.00)     Etc. 

This  last  question  relative  to  cost  is  an  inference.  From 
actually  seeing  that  6"  are  ^  of  13"  the  child  easily  infers 
that  $6.00  are  |  of  $12.00.  If  questions  were  asked  concern- 
ing time  required  to  build  the  fence  it  would  be  easy  for  the 
child  to  infer  that  6  days  are  ^  of  12  days; 


INTRODUCTION  H 

6  hours  are  ^  of  12  hours; 
6  men     are  ^  of  12  men. 
So  all  possible  combinations,  separations,  products  and 
parts  of  the  numbers  to  12  can  be  reached  through  the  use 
of  these  rulers. 

When  a  child  has  seen  that  the  sum  of  3"+3"=6"  and  has 
inferred  that 

3c  and  3c  =  6c ;  or 
3  yds.  and  3  yds.  =  6  yds. ; 
3  hours  and  3  hours  =  6  hours ; 
the  teacher  should  go  to  the  board  and  write  3 

3 


At  some  other  time,  other  than  the  construction  period, 
have  a  drill  on  writing  the  combination.  Write  it  as  a  whole. 
Erase  and  let  them  write  it.  AVrite  it  again  omitting  one  of 
the  numbers.  Erase.  Let  children  write  it  inserting  the  one 
the  teacher  omitted,  etc. 

Drills  of  this  sort  help  to  fix  number  relations  which 
have  been  previously  seen. 


MATHEMATICAL  CONSTRUCTION 

PLAN    OF   THE   LESSONS 
Folding 

I  have  in  the  folding  lessons  shown  what  variety  of  con- 
crete problems  can  be  given;  how  many  of  them  can  be 
correlated  to  the  finished  article;  how  they  do  incidentally 
show  the  number  relations  1,  2,  3,  4  and  more,  equal,  |,  ^,  ^. 

At  the  end  of  some  of  the  simple  folding  lessons,  I  have 
drawn  surfaces  which  the  child  sees  in  making  the  article. 
These  can  be  put  on  the  blackboard  and  concrete  problems 
based  on  them  after  the  folding  lesson. 

Folding  and  Free  Cutting 
(Weaving  and  Decorating) 

After  a  child  has  dealt  with  the  relations  1,  2,  3,  4,  equal, 
■J,  ^,  ^  in  folding  lessons,  he  is  ready  to  do  some  free  cutting. 

Weaving  and  Decorating  give  many  opportunities  for 
cutting  a  piece  of  paper  into  2  equal  pieces,  or  3,  or  4,  as  the 
case  may  be;  or,  for  cutting  a  piece  of  paper  twice  as  large, 
long,  or  wide  as  another. 

There  is  a  natural  sequence  of  the  three  topics  folding, 
free  cutting  and  measuring  in  the  single  subject  Weaving. 
The  weaving  lessons  are  presented  in  this  order.    No  concrete 

12 


PLAN  OF  THE  LESSONS  13 

problems  have  been  written  in  connection  with  these  lessons, 
but  they  should  be  given  wherever  there  are  surfaces  or  lines 
to  compare. 

These  woven  mats  can  be  utilized  in  many  ways.  They 
can  be  used  in  the  doll  house  for  mats,  table  cloths,  dresser 
scarfs,  sofa-pillow  tops;  they  make  a  good  background  for 
wall-pockets,  calendars,  match  scratchers;  woven  a  certain 
way  they  make  telephone  pads;  they  also  make  a  pretty  top 
for  blotters,  or  outer  side  of  napkin  rings  when  they  are 
under  transparent  celluloid.  Wall  paper  can  sometimes  be 
made  by  weaving.  Woven  mats  can  be  folded  and  pasted 
into  cornucopias,  lanterns,  and  all  sorts  of  baskets.  They 
can  be  used  to  decorate  books,  outside  of  banks,  envelopes,  etc. 

In  the  folding  and  free  cutting  weaving  lessons,  the 
development  of  the  relations  1,  2,  3,  4,  has  been  shown. 

In  the  measuring  lessons,  besides  showing  how  naturally 
and  definitely  number  relations  are  felt  in  using  these  single 
unit  rulers,  a  utility  has  been  given  for  every  mat.  For 
progressive  number  work,  however, -carefully  read  the  lessons 
given  under  the  heading  "Measuring  with  Single  Unit 
Rulers." 

Measuring  with   Single  Unit  Rulers 

In  measuring  with  the  Single  Unit  Rulers,  I  have  shown 
how,  through  a  series  of  lessons,  the  child  finds  all  combina- 
tions, separations,  products,  and  parts  of  numbers  up  to  12; 
also  the  great  variety  of  lines,  surfaces  and  solids  there  are 
to  compare;  how  I  correlate  my  concrete  problems  with  the 
finished  article. 

There  is  such  a  great  variety  of  these  surfaces  and  lines 
that  the  teacher  must  discriminate  in  comparing  them.    She 


14 


MATHEMATICAL  CONSTKUCTION 


must  not  ask  questions  on  every  possible  relation  she  sees; 
if  she  stops  too  often  the  child  will  lose  interest  in  what  he  is 
making.    That  is  worse  than  losing  the  number  work. 

In  making  an  article  based  on  a  square  consisting  of  4 
small  squares,  the  child  sees  the  relation  of 

1  to  3;  2  to  4;  3  to  6;  4  to  8;  5  to  10;  or  6  to  13. 


13. 


10 

4 

\ 

1 

<f':  -■ 


PLAN  OF  THE  LESSONS  15 

From  these  squares  the  following  articles  can  be  made: 
Booklet.  Picture  frame. 

Wall-pocket.  Salt  and  peppers. 

Soldier's  cap.  Boat. 

Bookmark.  Basket. 

Fireman's  cap.  Cup. 

Scissors  holder. 

In  making  an  article  based  on  a  square  consisting  of  9 
small  squares,  the  child  sees  the  relations  of  1,  2,  3 ;  2,  4,  6 ; 
3,  6,  9 ;  or  4,  8,  12.  From  9  squares  the  following  can  be 
made: 

Taboret.  Dutch  bonnet. 

Stove.  Wheel-barrow. 

Cube  bank.  Bos. 

Baskets  (cut  on  straight  folds  or  cut  on  diagonals). 

In  making  an  article  based  on  a  square  consisting  of  16 
small  squares,  the  child  observes  the  relations  of  1,  2,  3,  4; 
2,  4,  6,  8;  or  3,  6,  9,  12.  From  16  squares  the  following  can 
be  made : 

Baskets  (cut  on  straight  fold,  or  cut  on  diagonal). 


Cradle. 

Hallowe'en  lantern. 

Table. 

Pilgrim  cradle. 

Chair. 

Square  prism. 

Lounge. 

Square  pyramid. 

Bed. 

Cube. 

House. 

Wagons  and  carts  of  all  kinds. 

Wood  box. 

Sled. 

Box  with  lid. 

Pocket-book. 

16  MATHEMATICAL  CONSTEUCTION 

Dresser,  sectional  book  case,  or  any  kind  of  cabinet  with 
drawers  or  compartments. 

Then  there  are  all  sorts  of  odd  lengths  and  widths  of 
oblongs  to  be  used  as  sides  for  circular  baskets,  pails,  bowls; 
backgrounds  for  calendars,  match  scratchers,  wall-pockets; 
strips  and  mats  for  weaving.  These  give  such  combinations 
as: 

2  3         5         9         7 

3  4         4         3         3 


5        7        9       11       10  etc. 

The  following  chart  shows  every  combination  of  numbers 
from  1  to  12  inclusive,  and  the  lessons  in  which  the  child 
sees  these  combinations : 

All  the  Combinations  to  12  and  the  Lessons  in  Which 
They  are  Found 

Note.— These  are  in  the  measuring  section  unless  otherwise  specified, 

1     Lessons :     XIV,  XVII,  XX,  XXII,  XXVII. 
1 


2 


2     Lessons:     XIV,   XVII,   XIX,   XXII,   XXV,   XXVII, 
1  XXIX. 


PLAN  OF  THE  LESSONS  17 

3     Lessons :     VI,  XIV,  XVII,  XVIII,  XXVII,  VIII, 
1  (Weaving,  p.  78),  II  (Weaving,  p.  69),  XXIX. 


4    Lessons:     XXX,  XXI,  XXVII,  XXIX. 
1 


5     Lessons:  XXX,  X,  XXIV,  VII  (Weaving,  p.  76), 
1 


6 


6     Lessons:     XXIV,  XXXVII. 
1 


7  Lessons:  II  (Weaving,  p.  69),  VIII  (Weaving,  p.  78), 
1 

8 

8  Lesson:     XXXV. 
1 


18  MATHEMATICAL  CONSTEUCTION 

9     Lessons:  XXXII,  VII  (Weaving,  p.  76). 
1 

10 

10  Lesson:  VII  (Weaving,  p.  76). 
1 

11 

11  Lesson:     XXXV. 
1 

12 

2    Lessons:    IV,  XIII,  XVI,  XXVII,  XXXI,  VII,  XIV, 
2  XVII,  XXVIII,  XXXIV. 


3     Lessons :     V,  XXVI,  XXVII,  XXIX. 

2 


4    Lessons:     XXX,   VII,   XIII,   XVI,  XXVIII,   XXXI, 
2  XXXIV,  VI  (Weaving,  p.  74). 


5     Lessons:     XXX,  X,  XXXIII. 
2 


PLAN  OF  THE  LESSONS  19 

6  Lessons:     XIII,  XYI,  XXVIII,  XXXI,  XXXIV. 
2 

8 

7  Lesson :     XXX. 
2 


8  Lessons:  XXXI,  XXXIII,  XXXIV,  VI   (Weaving,  p. 

2  74). 

10 

9  Lesson:  VII    (Weaving,  p.  76). 
2 

11 

10    Lessons:     XXXIV,  XXXV. 
2 

12 

3  Lessons:     III,  VIII,  XII,  XXVIII,  XXXIII,  XXXVL 
3 

6 

4  Lessons:  VI,  VIII   (Weaving,  p.  78),  II   (Weaving  p. 
3  69). 


20  MATHEMATICAL  CONSTRUCTION 

5  Lesson :  •  XI. 
3 

8 

6  Lessons:     YIII,  XII,  XV,  XXY,  XXVIII,  XXXIII, 
3  XXXVI. 

9 

7  Lessons:    VI    (Weaving,   p.    74),    VIII    (Weaving,   p. 

3  78). 

10 

8  Lesson:  VIII  (Weaving  p.  78). 
3 

11 

9  Lessons :     XII,  XV,  XXVIII,  XXXIIL 
3 

12 

4  Lessons:     IX,  XIII,  XVI,  XXVIII,  XXXI,  XXXIV, 

4  XXXV. 

8 

5  Lessons:     XXX,  XXI,  XXXII,  XXXV. 

4 


9 


PLAX  OF  THE  LESSONS  21 

6  Lessons:  XXXI,  XXXIV,  VI  (Weaving,  p.  74). 
4 

10 

7  Lessons:  VIII  (Weaving,  p.  78). 
4 

11 

8  Lessons :     IX,  XXVIII,  XXXIV. 
4 

12 

5     Lessons:  XXVI,  VII  (Weaving,  p.  76). 
5 


10 


6     Lesson:  VII    (Weaving,  p.  76), 
5 


11 


7     Lesson :     X. 
5 


13 


6     Lessons:     XII,  XV,  XXVIII,  XXXIII,  XXXIV. 
6 


13 


22  MATHEMATICAL  CONSTEUCTION 

2,   1"— XIV,  XVII,  XXIX. 

2,  2"— IV,  VII,  XIII,  XIV,  XVI,  XVII,  XXVIII,  XXIX, 

XXXI,  XXXIV. 
2,  3"— III,    VIII,    XII,    XV,    XXV,    XXVI,    XXVIII. 

XXXIII. 
2,  4"— II,  IX,  XIII,  XVI,  XXVIII,  XXXI,  XXXIV. 
2,  5"— XXVI,  XXXV. 

2,  6''— XII,  XV,  XXVIII,  XXXIII,  XXXIV. 

3,  1''— XIV,  XXXVI. 

3,  2"— VII,  XIII,  XVI,  XXV,  XXVI,  XXVIII,  XXXI, 

XXXIV. 
3,  3"— VIII,  XII,  XV,  XXII,  XXV,  XXVIII,  XXXIII. 

3,  4"— IX,  XVIII,  XXVIII,  XXXII,  XXXIV. 

4,  1"— XIV,  XVII. 

4,  2"— XIII,  XVI,  XXVIII,  XXXI,  XXXIV. 

4,  3''— XII,  XV,  XVIII,  XIX,  XXVIII,  XXXIII. 

5,  1"— XXIX,  XXIV,  XXVII. 

5,  2"— XXVI,  XXXIV,  XXXV. 

6,  1"— XXXVI. 

6,  2"— X,  XIX,  XXVIII,  XXXIV. 


FOLDING 

LESSON  I 
Shawl 

Note. — Early  lessons  should  be  given  without  verbal  command.  Teacher 
should  only  show  what  to  do.     Then  let  children  imitate. 


Material :     An  oblong  piece  of  paper. 

Give  word  oblong  in  this  way : 
Show  me  a  long   edge  of  this  oblong. 
Show  me  a  short  edge  of  this  oblong. 
Show  me  an  edge  longer    than  a. 
Show  me  an  edge  shorter  than  h. 
Show  me  an  edge  shorter  than  c,  etc. 
Play  the  oblong  is  a  shawl. 

Which  edge  needs  more  fringe,  a  or  &?     (Teacher  points 
to  edges.) 

If  the  fringe  on  a  costs  a  nickel,  what  does  the  fringe  on 
h  cost?     (Ans.     More  than  a  nickel.) 

If  the  fringe  on  h  costs  a  nickel, 
what  does  the  fringe  on  a  cost?  (Ans. 
Less  than  a  nickel.) 

Teacher  folds  so  that   the  edge  a 
exactly  touches   &.      Children  imitate. 
Eesult:     Figure  2. 
Children  lay  theirs  on  desks.    Teacher  shows  large  undi- 
vided surface  to  them.     Play  that  it  is  a  piece  of  cloth  and 

23 


Fio.  a.. 


24 


MATHEMATICAL  CONSTRUCTION 


repeat  questions  on  relations  of  surfaces  as  before.  Viz. :  If 
it  takes  an  hour  to  hem  this  side  (pointing  to  d)  will  it  take 
more  or  less  to  hem  this  (pointing  to  c)  ?  Ans. .  More  than 
an  hour,  etc. 

Fold  back  oblong  and  tear  off. 

Eesult :     A  doll's  shawl. 

Surface  Seen  While  Making  Shawl 
Draw  them  on  the  blackboard  to  scale  of  6'''  to  V\ 


Suggestive  comparisons  on  which  to  base  problems: 

A=3  of  C.  E=2  of  C. 

=3  of  B.  =2  of  B. 

=sum  of  B+E.  =D. 

=sum,  of  D+B.  =sum  of  B+C. 

=sum  of  E+C. 

=8um  of  D+C. 


FOLDING 


25 


B=|  of  E. 
=i  of  D. 
=i  of  A. 


D=2  of  B. 
=2  of  C. 
=E. 
=A-B. 
=sum  of  B+C, 


Note. — If  child  does  not  see  the  relations  of  the  surfaces,  cut  pieces 
of  paper  the  exact  size  of  the  blackboard  figures.  Dissect  and  put  the 
pieces  together  in  any  necessary  form  to  give  the  child  a  clear  image  of 
the  relation  of  the  surfaces. 


LESSON  II 


Booklet 


Repeat  lesson  I  but  speak  of  the  oblong  being  a  cover  for 
the  library  table,  the  edges  of  which  are  stenciled,  hemmed, 
embroidered  or  fringed. 

When  A  is  obtained  the  children 
have  2  surfaces  each  a  triangle.  Play 
one  triangle  is  a  library  floor,  what  is 
opposite  surface?  (Ans.  Another 
floor  just  as  big.) 

If  it  takes  10  yds.  of  carpet  for  the  first  floor,  how  many 
yds.  for  opposite  floor?  The  teacher  just  points  to  the  sur- 
faces saying:  If  it  takes  10  yds.  for  this  floor,  how  many 
for  this?  (Ans.  Just  as  many,  10  yds.)  If  this  costs 
$25.00,  what  does  this  one  cost  ?  etc. 

Unfold  the  paper.     Eesult:     A  square. 


26 


MATHEMATICAL  CONSTRUCTION 


Play  this  is  a  rug  for  the  library. 

If  the  fringe  on  A  costs  a  dollar,  what  does  the  fringe  on 
C  cost?  on  D?  on  B?    How  many  dollars  for  fringe  on  A 
and    C   together?     D    and   B   together? 
II ow  many  dollars  for  all  sides? 

Fold  the  square  into  2  equal  oblongs. 
Here  are  two  oblong  surfaces  to  compare 
as  the  triangles  were  compared. 
Eesult:    A  booklet. 


LESSON  III 


Wall-pocket 
(New  relation  1  to  3) 

From  an  oblong  proceed  to  make  a  square  as  in  previous 
lessons,  but  do  not  ask  too  many  questions  on  the  surfaces 
and  lines  already  talked  about.  The  same  relations  with 
different  lengths  of  lines  and  size  of  surfaces  can  be  found 
in  the  new  folds. 

After  the  square  has  been  folded  into  2  equal  oblongs  the 
new  folds  begin. 

Fold  so  that  the  short  edges  touch. 

Eesult,  4  small  squares.  Children  see  only  the  outside 
ones. 

Teacher  points  to  1  square  and  says :  If  this  window 
glass  costs  a  dime,  what  does  this  one  cost?  (Pointing  to 
opposite  one.)      (Ans.     One  dime.) 

If  it  costs  a  quarter?     (Ans.     One  quarter.) 


FOLDING  27 

If  putty  on  one  edge  costs  a  penny,  what  does  putty  on 
another  cost?    What  does  putty  on  all  cost? 

Hold  square  so  that  4  free  points  point 
upward.     Teacher  moves  one  point  from  A  y/\. 

to  B,  thus  making  a  wall-pocket.  /^  \^ 

Now  child  sees  a  whole  triangular  sur- ^ A 

face   (C)   and  the  opposite  side  which  is  a     \^      c     / 
square.  \.     y"^ 

Now  play  C   is   a  piece  of  glass  for  a  ® 

window.  Teacher  points  to  opposite  surface  (the  square) 
and  says:  How  many  pieces  just  as  big  can  be  cut  from 
this?  (Ans.  2  pieces  of  glass.)  If  this  (pointing  to  C) 
costs  a  nickel,  what  does  this  (pointing  to  opposite  square) 
cost?     (Ans.     2  nickels.) 

If  it  costs  a  dime?     (Ans.    2  dimes.) 

If  it  costs  a  quarter?     (Ans.     2  quarters),  etc. 

Suggestive  Cojipaeisoxs  of  Surfaces  Seen  While  Making 
"Wall-pocket 

Draw  surfaces  on  blackboard  to  any  scale  large  enough  to 
be  easily  seen  by  pupils.     Base  problems  on  these  surfaces. 

A=sum  of  B+C.  =E 

=sum  of  D+E.  =2  G's 

=3  E's.  ^=^  ^'^• 

=3  F's  =*  of  ^• 

=3  B's.  =i  of  B. 

=6  G's.  =i  of  D. 

B=i  of  A.  =i  of  C. 

=1  of  C.  =h  of  A. 

=4  H's.  F=2  G-s. 

=4  of  D  =4  H's. 


28 


MATHEMATICAL  CONSTRUCTION 


E=C-B. 
=2  G's. 
=i  of  A. 
=^  of  D,  etc. 


i^ 


LESSON  IV 

Soldier's  Cap 


Made  just  like  wall-pocket  with  an  additional  fold  at  the 
end;  the  folding  back  of  the  three  points  which  were  left  as 
the  back  of  the  wall-pocket,    Result : 
Soldier's  cap. 

Number  relations  not  different 
than  Lesson  III;  but  this  should  be 
made  of  a  differently  sized  paper,  to 
give  a  new  length  of  lines  and  size 
of  surfaces  not  relative  but  actual. 


FOLDING  29 

Play  that  the  large  square  is  a  drill  ground  for  soldiers. 

If  the  ground  is  1  block  long,  how  wide  is  it?  (Ans.  1 
block  wide.) 

If  one  company  of  soldiers  can  stand  next  to  one  another 
on  this  side  (pointing  to  one  edge),  how  many  can  stand 
next  to  one  another  on  this  edge  (pointing  to  another  edge)  ? 
When  folded  into  oblongs  these  questions: 

If  there  is  room  for  one  company  to  drill  here  (showing 
1  oblong),  there  is  room  for  how  many  companies  to  drill 
here  (show  other  oblong)  ?  After  triangle  is  folded  back 
repeat  question  and  get  relation  1  to  2. 


LESSOI^  V 
Fireman's  Cap 

No  new  number  relation.  A  review  of  relations  found 
in  Lessons  III  and  lY.    Use  a  different  size  of  paper. 

Make  soldier's  cap;  then  press  the  front  and  back  points 
of  the  cap  together.  Hold  so  that  2  free  points  point  up- 
ward. Fold  one  of  them  back  until  it  touches  bottom  corner. 
Eesult :     Fireman's  cap. 

Play  that  the  oblongs  are  windows  in  the  engine  house. 
Question  on  time  it  takes  to  wash  them. 

Play  that  squares  are  floors.  Question  on  scrubbing.  Or 
play  that  squares  are  sheets  on  fireman's  bed.  Question  on 
hemming  of  edges;  or,  in  comparing  two  surfaces,  question 
on  length  of  time  to  iron  them. 


30 


MATHEMATICAL  CONSTRUCTION 


Play  that  triangle  is  a  piece  of 
rubber  for  making  the  hat.  Problems 
on  cost  of  it.  If  this  piece  of  rubber 
(pointing  to  A)  costs  a  nickel,  what 
does  this  cost  (pointing  to  opposite 
square)  ?  Ans.  2  nickels.  If  it  costs 
8  cents?    Ans.    2,  8  cents. 


LESSON  VI 

Envelope 
(New  relation  1  to  4  and  1  to  3.) 

Play  the  square  is  the  top  of  a  desk. 

Compare  cost  of  varnishing  to  another  child's  square  just 
the  same  size. 

When  folded  on  a  diagonal,  play  that  one  triangle  thus 
formed  is  a  tile  of  the  school  vestibule  floor.  Compare  cost 
of  this  with  opposite  triangla 

Teacher  holds  her  paper 
so  that  folded  edge  is  down. 
She  folds  one  of  the  upper 
points  (A)  to  center  of  folded 
edge.  Children  imitate.  Ee- 
sult:    Fig.  1. 

5  is  a  whole  surface  itself. 
Compare  it  with  opposite  surface.    Kelation  1  to  4. 


FOLDING 


31 


If  B  is  one  tile  opposite  surface 
is  4  tiles. 

If  B  costs  a  dime  opposite  sur- 
face costs  4  dimes,  etc. 

Fold  point  C  to  E.  Result: 
Fig.  2. 

F  is  a  whole  surface,  so  is  G; 
compare  F  to  G.     Relation  1  to  2. 


FI3-2. 


Fold  point  D  (Fig.  1)  to  ^.    Result:    Fig.  3. 

Compare  F  and  H.     Relation  equal. 
Compare  F  and  G.     Relation  1  to  2. 
Compare  H  and  G.     Relation  1  to  2. 
Compare  G  with  opposite  surface.     Re- 
lation 1  to  3. 

Of  course,  give  concrete  problems  to  get 
pjg  3  these  relations. 


Surfaces  Seen  "While  Making  Envelope 
Draw  them  enlarged  on  blackboard. 


32 


MATHEMATICAL  CONSTRUCTION 


Base  problems  on  some  of  these  comparisons; 


A=2  of  B. 

G  =8  of  E. 

=2  of  G. 

=4  of  D. 

=8  of  D. 

=i  of  A. 

=8  of  H. 

K=3  Ws. 

C=3  of  D. 

=F-D. 

=6  of  E. 

=6  of  J. 

=3  of  H. 

=sum  of  H,  D,  E,  J. 

=sum  of  K,  E,  J. 

E^  of  D. 
=i  of  C. 
=i  of  F. 
=i   of  K. 

B=sum  of  C+D. 

D=^  of  F. 
=H. 

G=B. 

=2  of  J. 

=4  of  H. 

=i   of  C. 

=8  of  J. 

=i   of  B. 

LESSON  VII 


PiCTUKE  Frame 

Relation  1  to  4;  1  to  5;  1  to  6. 

Fold  square  into  2  equal  oblongs. 

Fold  so  that  short  edges  of  oblongs 
touch.  Open.  Large  square  is  now 
divided  into  4  small  squares.     (Fig.  1.) 


E      .. 


fy./. 


FOLDING 


33 


Fold  so  that  point  A  touches  center 
(E). 

Fold  so  that  point  B  also  touches  center 
(E.)   (Eesult,  Fig.  2.) 

Compare  surface  F  with  surface  G,  Fig. 
2.    Relation  equal. 

Compare  surface  F  with  opposite  sur- 
face.   Eelation  1  to  6. 

Compare  surface  G  with  opposite  sur- 
face.   Eelation  1  to  6. 

Fold  so  that  point  D  touches  center  E, 
Fig.  3. 

Compare  H  to  F  or  G.     Eelation  equal. 

Compare  H  to  opposite  surface.  Eela- 
tion 1  to  5. 

Fold  point  C  to  center  E,  Eesult,  Fig.  4 
Compare  surface  J  to  opposite  surface. 
Eelation  1  to  4. 

Fold  back  point  1  to  middle  of  edge  5. 

Fold  hack  point  3  to  middle  of  edge  6. 

Compare  K  and  M  (Fig.  5). 

Compare  K  and  H,  etc. 

Fold  back  point  4  to  8  and  3  to  7. 

Result:    Picture  frame  for  a  doll's  house. 

Base  the  concrete  problems  on  things 
pertaining  to  a  doll's  house. 


^E 

I 

D  '  E 


Vi<f.a. 


r/?3 


T.qS 


34 


MATHEMATICAL  CONSTRUCTION 


The  squares  and  oblongs  can  be  lace  curtains  for  the  doll 
house.  Question  about  length  or  cost  of  lace  for  the  curtains. 
Some  curtains  having  lace  on  2  edges,  others  on  3  edges. 

Let  triangle  F  (Fig.  4)  be  a  doll's  shawl.  Let  opposite 
surface  be  a  piece  of  cloth.  Ask  how  many  shawls  can  be  cut 
from  it. 

Let  small  square  (Fig.  1  before  it  is  opened)  be  a  hand- 
kerchief. Question  on  cost  of  lace  on  edges,  or  time  to  hem 
them. 

Surfaces  Seen  While  Making  Picture  Frame 

Draw  them  enlarged  on  blackboard.  Suggestive  compari- 
sons : 


A=2  of  B. 
=8  of  C. 
=2  of  I. 


A=sum  of  D+C. 
=sum  of  H+3  C's 
=sum  of  1+4  C's. 


FOLDING 

A=sum  of  G+2  C's. 

D=B+3  C's 

=sum  of  D+E+F. 

=7  C's. 

=sum  of  G+C+E+F. 

G=6  C's. 

C=i  of  B. 

=1+2  C's. 

=i  of  I. 

H=I+C. 

=siim  of  E+F. 

=5  C's. 

B=sum  of  C+E+F. 

1=4  C's. 

=1  of  A. 

F=3  E's. 

=1. 

=C-E. 

35 


LESSOX  VIII 


Salt  and  Pepper  Holder 


Fold  square  on  its  diagonals. 

Fold  points  a,  h,  c,  d,  to  center  E  as 
in  previous  lesson,  getting  small  tri- 
angles to  compare  with  opposite  surface. 
Eelations  7,  6,  5,  -i,  respectively.  Turn 
Fig.  1  upside  down. 


Fig.  1. 

Fold  the  four  points  of  the  new 
square  to  center  E  again  and  turn  up- 
side down. 

Insert  fingers  under  points  1,  2,  3,  -i. 
Fig.  2,  and  squeeze  into  the  shape  of 
salt  and  pepper  holder.    Fig.  3. 


I     2. 
3  ¥ 


Fig.  2. 


36  MATHEMATICAL  CONSTEUCTION 

After  the  holder  is  made,  inquire: 
If  one  little  pocket  holds  Ic  worth  of 
salt,   how   many   cents'    worth   will    all 
hold  ? 
i,-i„   -  If  it  holds  2c  worth,  how  many  2c 

worth    will    all    hold?      (Ans.      4,    2c 
worth.) 


LESSON  IX 

Boat 

See  Lesson  IV,  Soldier's  Cap,  for  beginning  of  boat. 
Take  the  two  side  points  of  the  soldier  hat  and  make  them 
touch  one  another.  Now  there  are  2  free  points  of  a  square. 
Fold  one  of  them  back  to  opposite  corner.  Fold  the  other 
one  back  the  other  way.  Eesult :  A  little  soldier's  hat. 
Press  the  two  free  points  of  this  together. 
Fold  back  both  in  opposite  directions  to  meet  opposite 
corner.  Eesult:  A  very  small  soldier's  cap,  with  3  points  at 
the  top.  Put  fingers  on  sides  of  middle  point  and  pull  the 
paper.    Eesult:  Boat. 

When  boat  is  made  ask  questions  about  the  prow  and 
stern  of  it.    Bind  them  with  iron  (imaginatively,  of  course). 
~7''\  ~?  If  prow  is  5  yds.  long,  stern  is 

'     \    /y        5  yds.  long. 

1/  If  it  took  2  days  to  bind  prow 

with  iron,  it  took  2  days  to  bind 
stern  with  iron. 

If  iron  on  prow  costs  $10.00,  iron  on  stern  costs  $10.00. 


FOLDING  37 

If  it  takes  a  week  to  paint  1  side  of  ship,  it  takes  a  week 
to  paint  other  side  of  it. 

If  it  takes  $100.00  worth  of  wood  to  build  one  side  it 
takes  $100.00  worth  to  build  other  side. 

Compare  cost  of  painting  1  side  of  cabin  to  1  side  of  ship 
on  the  outside.  (Eelation  of  1  side  of  cabin  to  side  of  ship 
is^). 

Compare  cost  of  painting  1  side  of  cabin  outside  and 
inside  to  1  side  of  ship  outside  and  inside,  etc. 


LESSON  X 

Square  Prism 

(To  be  used  for  that  game  in  which  the  teacher  says : 
"I  am  thinking  of  something  which  looks  like  a  square  prism. 
Guess  what  it  is.'") 

Fold  a  square  into  two  equal  oblongs.  Open.  (Fig.  1.) 
Make  edge  A  touch  folded  line.  Compare  oblong  thus 
obtained  to  opposite  surface.  Rela- 
tion 1  to  3.  Make  edge  B  touch  folded 
line.  Compare  oblong  thus  formed  to 
opposite  surface.  Relation  1  to  2. 
Open  whole  paper  and  fold  the  other 
way  into  2  equal  oblongs. 

Fold  long  free  edges  back  to  long 
'^  '  folded  edge.     Result:    Large  square  is 

divided  into  16  small  squares. 


38 


MATHEMATICAL  CONSTRUCTION 


A 

B 

C 

D 

E 

F 

G 

H 

S       V- 


Cut  line  1  and  fold  back  whole  upper  row  of  squares. 
(Fig.    2).      Compare   surface   A    with 
surface  bed.    Eelation  1  to  3. 

Cut  line  2. 

Compare  surface  a  with  surface  cd. 

Compare  surface  cd  with  opposite 
surface.     Eelation  1  to  6.     Cut  line  3. 

Cut  line  4,  fold  back  whole  lower 
row.  Compare  surface  ef  with  sur- 
face gh. 

Compare  surface  ef  with  opposite  surface.   Eelation  1  to  4. 

Compare  surface  gh  with  oj)posite  surface.  Eelation  1  to  4. 

Cut  line  5.  Compare  surface  e  with  opposite  surface. 
Eelation  1  to  8. 

Cut  line  6. 

Fold  and  paste  into  square  prism  shape. 

Play  game  mentioned. 

Concrete  problems:  In  Fig.  1,  after  a  is  folded  to  middle 
fold  ask: 

"If  this  wall  (small  oblong)  needs  two  rolls  of  paper  to 
paper  it,  how  many  rolls  will  this  wall  (opposite  surface) 
need?"    (Ans.  3,  2  rolls.) 

After  B  is  folded  to  middle  fold  ask :  ''If  one  man  paints 
this  wall  in  4  hrs.,  how  long  will  it  take  him  to  paint  this 
one  (opposite  surface)  ?"  (Ans.  2,  4  hrs.) 

Children  should  keep  square  prism  in  school  for  future 
comparison  with  the  cube  which  is  to  be  made  next  out  of 
the  same  sized  paper,  as  square  prism  was  made. 


FOLDING 


39 


Surfaces  Seen  While  Making  Square  Prism 

Enlarge  and  draw  them  on  the  blackboard. 

Base  concrete  problems  on  some  of  these  comparisons: 


H 


A=2  of  B. 

=G+H. 

=4  of  C. 

=A-D. 

=  8  of  G 

=D-B. 

=sum  of  D+C. 

=2  times 

G. 

=sum  of 

B+2  Cs. 

=4  times 

E. 

B=2  of  C. 

F=3  times 

E. 

=D-C. 

=sum  of 

G+ 

=A-2  Cs. 

C=i  of  A. 

=A-B. 

=^  of  B. 

B=sum  of  C+E+F. 

F=C-E. 

=sum  of 

C+G+H. 

=i  of  D. 

=sum  of 

E+F+G+H. 

G=2  Es. 

=4  of  G. 

=H. 

=^  of  A. 

=F-E. 

C=^  of  D. 

=C-H. 

=F+E. 

40 


MATHEMATICAL  CONSTEUCTION 


1 

A     B 

2. 

G 

3 

E 

F 

15        If       jfc 
•          •          1 

LESSON  XI 

Cube 

Fold  16  squares  as  in  previous  lesson.     Cut  off  lower  row. 
Compare  this  to  surface  which  is  left.     If  large  surface  will 
hold  a  loaf  of  cake,  what  will  smaller 
surface  hold?     (Ans.  ^  of  a  loaf.) 

If  it  takes  3  cups  of  flour  for  the 
former  what  will  it  take  for  the  latter  ? 
(Ans.  -J  of  3  cups). 

Fold  back  row  of  3  squares.    Com- 
pare this  to  opposite  surface  in  same 
way  getting  result  ^.     Then  put  ques- 
tions the  opposite  way,  getting  result  3. 

Fold  back  other  outside  row  of  3  squares  and  repeat  ques- 
tions, getting  answers  ^  and  2. 

Open    folds,    making   large   flat   surface   with    12    small 
squares.    Make  cut  1.    Fold  back  surface  hcd. 

Compare  it  with  opposite  surface.     Eelation  ^  and  3. 
Make  cut  2.     Compare  cd  with  opposite  surface. 
Through  questions  get  relation  ^  and  4. 
Make  cut  3. 

Make  cut  4.     Fold  back  ef.     Compare  with  opposite  sur- 
face.   Eelation  ^  and  3. 
Make  cut  5. 
Make  cut  6. 
Fold  and  paste  into  a  cube. 


Compare  Cube  axd  Square  Prism 

How  many  cubes  like  this  can  be  cut  from  the  square 
prism  ? 


FOLDING 


41 


How  many  times  larger  is  the  square  prism  than  the  cube  ? 

The  cube  is  what  part  of  the  square  prism?  If  the  cube 
is  a  child's  building  block,  how  many  can  be  made  from  the 
square  prism?  If  the  little  block  costs  a  nickel,  what  doea 
the  big  one  cost?  (Ans.  2  nickels).  If  the  prism  costs  6c 
what  does  the  cube  cost?     (Ans.    ^  of  6c),  etc. 

Surfaces  Seen  While  Making  Cube 

Enlarge  and  draw  them  on  the  blackboard. 

Base  concrete  problems  on  some  of  these  comparisons: 


J 


C=^  of  A. 
=^  of  B. 


C=^  of  D. 
=i  of    I. 


42 


MATHEMATICAL  CONSTKUCTION 


C=2  of  K. 
E=J  of  F. 

=1  of  G. 

=i  of  H. 

=^  of   J. 

4  of  D. 

=F-G. 

=D-F. 
H=sum  of  G+E. 

=sum  of    J+E. 


D=sura  of  H+E. 
I=B. 

=2  of  C. 

=1  of  A. 

=4  of  K. 

=sum  of  J+K. 
J=3  of  K. 

=G. 

=2  of  E. 

=C+K. 


LESSON  XII 


Cradle 


Cut  ofE  lower  row  of  squares.  Compare  this  with  large 
surface.  If  small  surface  is  enough 
cloth  for  a  pillow  case  for  baby's  bed, 
what  is  large  surface?  (Ans.  Enough 
for  3  pillow  cases).  Make  cut  1. 
Fold  back  h-c-d.  Compare  it  with 
large  surface.  Eelation  1  to  3.  Fold- 
ing back  -  a-e-f  and  comparing  it  to 
what  is  left  on  opposite  side  gives  the 
relation  1  to  2.  Make  cuts  2,  3  and  4. 
Fold  and  paste  into  box  shape.  Eock- 
ers  to  be  made  free-hand  from  row  of 
squares  cut  off  originally,  then  pasted 
on. 


A        B    1    0     1   D 
E    !          I          ' 

'        1         ' 

1         ' 

1        1         ' 

^ 


FOLDING 


43 


New  Surfaces  Seen  "While  Making  Cradle 

(For  other  surfaces  see  lesson  on  Cube). 
Enlarge  and  draw  them  on  the  blackboard.    Base  concrete 
problems  on  the  comparisons : 


A 

B 

A=i  of  1). 

B  =  3  of  A. 

D=2  of  A. 

=i  of  B. 

:=3  of  C. 

=3  of  C. 

=C. 

C=l  of  D. 

=sum  of  A  +  C 

B=sum  of  D+C. 

=i  of  B. 

=B-C. 

=sum  of  D+A. 

=B-D. 

=B-A. 

LESSON  VIII 

Chair 

Fold  16  squares.  Cut  off  one  row.  Make  cut  1.  Fold 
back  ah.  Compare  this  with  opposite  surface.  Play  it  is  a 
blackboard.  Pointing  to  opposite 
surface,  this  blackboard  is  how  many 
times  .'as  large?  (5  times.)  If  it 
takes  one  boy  1  minute  to  clean  it, 
how  long  will  it  take  him  to  clean  the 
other?  (5  minutes.)  How  many 
boys  could  clean  it  in  1  minute?  (5  boys.)  Make  cut  2. 
Fold  back  c-d  and  repeat  comparisons — getting  result  4  in- 


44 


MATHEMATICAL  CONSTBUCTION 


stead  of  5.  Make  cuts  3  and  4.  Fold  back  e  and  /.  Com- 
pare c-d  to  what  is  left  on  opposite  side.  Result :  Relation  3. 
By  reversing  questions  get  relations  1/5,  1/4,  1/3,  respectively. 

Paste  e  under  h. 

Paste  /  under  g. 

Paste  a  under  g  and  /. 

Paste  c-d  on  i-j  to  strengthen  back.  Then  chair  is  made. 
Legs  may  be  cut  out  free-hand. 


A 

B 

d 


Chair 

Enlarge  and  draw  on  blackboard 
new  surfaces  not  seen  in  previous  les- 
sons. 

Give  concrete  problems  based  on 
these  comparisons: 

A=l/5  of  B. 

-i  of  C. 

=1  of  D. 
B=sum  of  C+A. 
=sum  of  D+2  A's. 
C=4  of  A. 

=B-A. 

=D+A. 
D=3  of  A. 

=C-A. 
=B-8  A's. 


FOLDING 


45 


^ 


LESSON  XIV 
Buggy 


IJL 
I 


\H- 


Prom  two  equal  oblong  papers  fold  to  get  squares.  From 
the  little  extra  oblongs  cut  wheels  and  shafts  free-hand. 

Fold  one  square  into   16   small 
squares.     Cut  off  1  row. 

Compare     this    to     13     squares 
which  are  left. 

Base    problems    on   things   seen 
when  driving. 

If  a  requires  10c  worth  of  grass 
seed,  &  requires  3,  10c  worth. 

If  it  takes  2  days  to  put  a  cement  walk  on  a,  it  takes  3, 
3  days  to  put  one  on  &. 

If  a  yields  5  bu.  corn  h  yields  three  5  bu. 

If  &  yields  3  pk.  potatoes  a  yields  ^  of  3  pk.,  etc. 

Make  cuts  1  and  2.  Fold  back  left  row  and  compare  to 
opposite  surface.     Relation  1  to  3. 

Make  cuts  3  and  4.  Fold  back  right  row  and  compare 
to  left  row  (equal).  Also  compare  to  opposite  surface.  (Eela- 
tion  1  to  2.)  Fold  and  paste  into  box  shape.  Do  same  witli 
other  square. 


46 


MATHEMATICAL  CONSTRUCTION 


If  one  holds  ^ 


pk 


Compare  volumes  of  two  box  shapes, 
oats  for  liorse  the  other  holds  |  pk.,  etc. 

Cut  one  of  these  as  indicated  in  Fig.  2. 

Insert  and  paste  this  into  other  box  shape.    Use  toothpicks 
for  axles,  adjust  wheels  and  paste  on  the  shafts. 


BASKETS 

This  is  a  series  of  basket  patterns  more  than  anj^thing 
else,  just  to  show  what  a  variety  can  be  made  from  16  squares 
cutting  either  on  a  straiglit  fold  or  a  diagonal.  I  have  par- 
tially shown  the  variety  of  surfaces  to  be  compared,  but  have 
not  inserted  any  concrete  problems,  though  of  course  they 
should  be  given  whenever  any  comparing  is  done. 


1 
1_  _ 

1 

« 

1       • 
_i J 

1 

I 

1 
1       , 

1 

1 
1 

I 

I         I        I 

'•II 

L_i-.a--'-  -J 


LESSON  XV 

Ordinary  box  shape,  based  on   16 
squares,  straight  cuts. 


LESSON  XVI 

Oblong  box  shape.  1  row  of  squares 
cut  off,  straight  cuts.  Cutting  for  flaps 
different  to  give  differently  shaped  sur- 
faces on  which  to  base  problems. 


FOLDING 


47 


,  -       J 

'        I        , 

'        I        I 

;    :    : 

I 

!    ■    :    1 


LESSON  XVII 


16  squares;  2  rows  cut  off;  straight 


cuts. 


Fig.  3  is  the  result 
when  the  free  corners  are 
cut  off. 


LESSON  XVIIL 


— ^ 

'               -               1 

1 — ' — ..-1 

1           '          • 

1 
.  — . 

1        ' 

L._J 

Like  a  cube 
with  two  sides 
cut  out. 


48 


MATHEMATICAL  CONSTRUCTION 


LESSON  XIX 


Box  shape;  9  squares;  straight  cuts.     When  bottom  row 
is  cut  oS  number  relation  is  1  to  3. 

When   right   row   is   cut   off  number 
relation  is  1  to  3. 

Make  cuts  a  b.  Fold  back  2,  3,  and  1. 
The  sum  of  2  and  3  can  be  cut  how  many 
times  out  of  opposite  surface?  Sum  of 
2  and  3  is  what  part  of  opposite  surface? 
Make  cut  C  and  fold  back  4,  Compare  to 
opposite  surface.  Eelation  1  to  5.  Make  cut  d. 
Fold  and  paste  into  box  shape. 


LESSON  XX 


• 

1 

• 

\ 

I 

\f  . 

N 

1 

/ 

>. 

• 

2 

/     ;\ 

/ 

u  _  N 

\ 

H 

1      • 

\ 

. 

\/ 

\ 

• 

1 

% 

1                                    N 

u___ 

.>*/. 

1 

Make  cuts 
indicated  and 
fold  on  the 
d  iagonals 
seen.  Paste  1 
on  2.  Make 
handle  of  3 
strips  cut 
free  -  hand. 
Result : 
Fig.  2. 


FOLDING 


49 


LESSON  XXI 


Eight  squares.  Di- 
agonal cuts. 

Cut  off  free  cor- 
ners if  you  so  desire. 


LESSON  XXII 


Q 

E 

!\ 

1 

1 
1 

/: 

rA 

^ 

Q. 

-i^ 

■^ 

2/ 

1 

1 

\i 

H      p 


Six  squares.  Diagonal  cuts.  After 
a  and  h  are  cut  out,  the  sum  of  a  and 
h=^  of  whole  surface  left. 

a  is  ^  of  whole  surface. 

&  is  :^  of  whole  surface. 


r^ 


Whole  surface=4  of  a  or 

4  of  6  or 

2  times  the  sum  of  a  and  b. 
Fold  on  horizontal  dotted  line. 
Now  a  is  ^  of  surface  seen. 
Now  &  is  ^  of  surface  seen. 
Sum  of  a  and  h  equals  surface  seen. 
Cut  a  on  horizontal  dotted  line. 
Triangle  1  is  4  of  h. 
B=2  of  triangle  1. 


Open. 

Fold  back  on  line  e-f. 


50 


MATHEMATICAL  CONSTKUCTION 


Compare  h  to  surface  seen  (^). 
Fold  on  horizontal  dotted  line  again. 
Compare  triangle  1  to  surface  seen  (^). 
Open. 

Fold  on  lines  g-h  and  e-f. 
Compare  h  to  surface  seen  (^). 
Compare  sum  of  A  1  and  A  2  to  surface  seen  (^). 
Compare  sum  of  A   1  and  A  2  and  h  to  surface  seen 
equal. 

Fold  and  paste. 


LESSON  XXIII 

Fold  back  whole  upper  row  after  cuts  1,  2  are  made. 
Compare  Fig.  a-h-c  with  opposite  surface.     Relation  1  to  3. 

Make  cut  3.     Fold  -back  whole  left  row. 

Compare  surface  a-h-c  to  surface  d-e-f.  Compare  d-e-f 
to  opposite  surface.  Relation  1  to  2.  Make  cut  4.  Compare 
g-h-i  to  opposite  surface.    They  are  equal. 

Make  long  edge  of  triangle  /  touch  right  edge  of  triangle 
g  and  paste.    Treat  other  corners  similarly. 

Cut  two  long  strips  free-hand  for  handles. 


z 

B  ;  c/ 

E  ' 

1 

«!^' 

3 

1 

1 
1 

^ 

L  .   i 

_  J.  - . 

_ 

-  4 


FOLDING 


51 


LESSOX  XXIV 


l\          «         t/Li 
-  — -^'j 1 -f  -- 

^  ;      I      [  ^ 

..:..Lj._ 

Make  cuts  1  and  2. 

Fold  back  surface  a-h-c-d. 

Compare  this  with  opposite  sur- 
face. 

Eelation  1  to  3. 

Make  cuts  3,  4. 

Fold  back  surface  e-f-g-h. 

Compare  this  surface  to  surface 
a-h-c-d  (equal). 

Compare  it  to  opposite  surface  (relation  1  to  2). 

Fold  back  i-j-h. 

Compare  i-j-k  to  opposite  surface  (relation  1  to  2). 

Fold  back  l-m-n. 

Compare  it  to  i-j-k. 

Compare  it  to  opposite  surface  (equal). 

Make  long  edge  of  A  a  touch  lower  edge  of  A  i.  Paste. 
Treat  other  corners  similarly.  Cut  ofE  points  which  pro- 
trude.   Cut  handle  free-hand. 

This  makes  a  nice  basket  for  Red  Riding  Hood,  or  a 
shopping  basket. 


^rr? 


52  MATHEMATICAL  CONSTRUCTION 

LESSON  XXV 

Make  cuts  1,  2. 
Fold  back  whole  upper  row. 
Compare  a-h-c-d  to  whole  opposite 
surface  (|). 
Cut  3,  4. 
Fold  back  e-f-g-h. 

Compare  surface  a-h-c-d  to  surface 
e-f-g-h  (equal). 

Compare  i  to  surface  a-h-c-d. 
Compare  i  to  /. 

Compare  sum  of  i  and  ;'  to  surface  a-h-c-d.     (^). 
Compare  e-f-g-h  to  opposite  surface.      (Eemember  that 
top  row  is  turned  back).    Relation  1  to  3. 
Fold  back  i-k-l-m;  also  o. 
Compare  irl--l-m  to  a-h-c-d;  or  e-f-g-h  (equal). 
Compare  i-h-l-m  to  opposite  surface  (^). 
Fold  and  paste  so  that  longest  edge  of  triangle  a  touches 
lower  edge  of  triangle  i.    Treat  other  corners  in  like  manner. 
Cut  off  protruding  points.     Eesult:  workbasket  for  sewing 
materials. 


FOLDING  AND  WEAVING 


LESSON  I 

Material:     Paper  6"x9". 

Fold  and  tear  so  that  a  6-ineh  square  is  obtained.     Color 
remaining  oblong,  which  is  6"x3",  on  one  side  with  crayola. 
Compare  the  two  surfaces,  giving  problems  on  size  and 
cost  of  rugs;  also  fringe  on  edge.     Fold  6  inch  square  into 
16  small  squares.    Then  fold  into  two  equal  oblongs. 
Make  cuts  indicat- 
ed at  a  and  h  on  fold- 
ed edge.  Fig.  1.  Open. 
Weave   with   the    col- 
ored    strip.      Eesult : 
Fig.  2. 


B 


I        <        ' 

•         I        • 

1 J I 


LESSOX  II 


"—■—'—■— *—^i^ 


B 


Eepeat  Lesson  I,  but  tear  off 
one  row  of  the  sixteen  squares 
making  cuts  a-h  (Fig.  1)  on 
folded  edge. 

Weave  with  colored  strip. 

Result:    Fig.  2. 


53 


54 


MATHEMATICAL  CONSTRUCTION 

LESSON  III 


Paper  same  as  Lesson  I. 

Eold  and  tear  to  get  6-inch  square.  Color  with  crayola 
the  oblong  which  is  left.  Then  divide  it  into  two  equal 
oblongs  by  folding  and  cutting  or  free  cutting. 

Fold  6-inch  square  into  16  small  squares.     Open. 
Fold  into  2  equal  oblongs. 

Make  cuts  indicat- 
ed at  a,  h,  c  on  folded 
edge  (Fig.  1).  Open 
and  weave.  Result : 
Fiff.  2. 


" — : i f 




LESSON  IV 

Paper  6"x9''. 

Make  6-inch  square  and  color  oblong  which  is  left.  Cut 
this  oblong  into  4  equal  oblongs  by  folding  and  cutting  on 
fold  or  free  cutting.  Fold  square  into  16  small  squares. 
Open.     Fold  into  2  equal  oblongs. 

Make  cuts  indicat- 
ed at  a,  b,  c  on  folded 
edge  (Fig.  1).  Open 
and  weave.  Eesult : 
Fi.o:.  2. 


iffil 


FOLDING  AND  WEAVING 


55 


LESSON  V 

Paper  6"x9". 

Fold  so  that  a  6-ineh  square  is  obtained.    Color  the  oblong 
which  is  left.     Divide  it  into  2  equal  oblongs.    Fold  square 
into  16  small  squares.    Open  and  fold  into  two  equal  oblongs. 
Make   cuts   indicated    at   a^   1),   c   on 
folded  edge    (Fig.   1).     Xow  with  free- 
hand cutting  (judging  with  the  eye,  there 
is  no  fold  to  cut  on) 
make  cuts  d,  e.    Open 
and    weave.      Eesult : 
Fiff.  2. 


ffi 


LESSON  VI 

Paper  6''x9". 

Fold  so  that  a  6-inch  square  is  obtained.  Color  oblong 
which  is  left.  Cut  it  into  4  equal  oblongs.  Fold  square  into 
16  small  squares.  Open  and  fold  into  2  equal  oblongs.  Make 
cuts  indicated  at  a,  b,  c,  on  folded  edge. 

Eesult:    Fig.  1. 

Make  cuts  d  and  e  free-hand.  Weave. 
Result:     Fig.  2. 


/I    O    6    E  C 


Color  the  border  if  you  care  to. 


56 


MATHEMATICAL  CONSTRUCTION 
LESSON  VII 


Two  8-inch  squares  contrasting  colors. 
Fold  the  one  which  is  to  be  used  for  strips  into  8  equal 
oblongs.    Cut  on  folds.    Fold  the  other  square  into  8  oblongs 
one  way.     Open  and  fold  into  8  oblongs  the  other  way. 
Open  and  fold  into 
2  large  equal  oblongs.   _  [ 
Make    cuts    indicated 
in    Fig.    1    on   folded 
edge.  Open  and  weave. 
Eesult:    Fig.  2. 


LESSON  YIII 


Two  8-inch  squares  contrasting  colors. 

Fold  the  one  to  be  used  for  strips  into  4  equal  oblongs. 
Cut.  Then  divide  one  of  those  oblongs 
into  2  equal  smaller  oblongs   by  free 


cutting  or  folding. 


Fold  the  other  8-inch  square  into 
4  equal  oblongs. 

Fold  the  two  free  edges  back  to  the 
opposite  folded  edge.     Open. 


Fold  the  other  way  in  same  manner.  Make  cuts  and 
weave  with  a.  narrow  strip  first,  two  wide  ones  next  and  a 
narrow  one  last.    Eesult :  Fig.  1. 


FOLDING  AND  WEAVING 


67 


LESSON  IX 


Two  8-inch  squares  contrasting  colors. 

Fold  the  one  to  be  used  for  strips  into  4  equal  oblongs. 
Divide  2  of  these  lengthwise  into   2   equal  oblongs  by  free 
cutting  or  folding. 

Fold  the  other  8-inch  square  into  4 
equal  oblongs.  Fold  the  2  free  edges 
back  to  meet  opposite  folded  edge.  Open 
and  fold  the  other  way  in  same  manner. 
Make  cuts  and  weave  with  2  narrow 
strips  first,  then  a  wide  one,  then  two 
more  narrow  strips.    Kesult:  Fig.  1. 


m 

— L_j — LJ L-j — I — I 


LESSON  X 


Two  squares,  contrasting  colors,  any  size  that  is  large 
enough. 

Fold  the  one  to  be  used  for  strips  into  2  equal  oblongs. 
Divide  one  of  these  into  2  smaller  oblongs.  Divide  one  of 
these  2  small  ones  into  two  equal  smaller  oblongs.     Cut. 

Fold  the  other  square  into  4  equal  oblongs. 

Fold  the  2  free  edges  back  to  oppo- 
site folded  edge.  Open  and  fold  the 
other  way  in  same  manner.  Make  cuts, 
open  and  weave,  using  two  narrow  strips 
first,  then  the  wide  one,  then  the  other 
two  narrow  ones  ;  going  over  one  and  un- 
der two.    Eesult :  Fiff.  1. 


58 


MATHEMATICAL  CONSTEUCTION 


LESSON  XI 


Two  squares,  contrasting  colors. 

Cut  the  square  to  be  used  as  strips  into  4  equal  oblongs. 
Bisect  two  of  tliese  lengthwise  and  then  divide  2  of  the  nar- 
row oblongs  thus  obtained  into  2  equal  smaller  ones. 

Fold  the  other  square  into  8  equal 
oblongs.  Open  and  fold  the  other  way 
into  4  equal  oblongs.  Fold  back  the  2 
free  edges  to  the  opposite  folded  edge. 
Open  so  that  the  paper  is  folded  into 
halves. 


Make  cuts  indicated  from  folded  edge 
(Fig.  1).  Open  and  weave,  going  over 
two  and  under  one,  using  one  middle 
sized  strip  first;  then  2  narrow  ones,  then 
1  wide  one,  then  2  narrow  ones,  then  1 
middle  sized  one.     Eesult :     Fig.  2. 


r '  !  i  1  :  i  ' 


LESSON  XII 


Two  squares,  contrasting  colors. 

Cut  the  square  to  be  used  as  strips  into  4  equal  oblongs 
Divide  each  one  of  these  lengthwise  into  4  equal  oblongs 
Sixteen  narrow  strips  are  the  result. 

Fold  the  other  square  into  8  equal 
oblongs.  Open  and  fold  the  other  way 
into  4  equal  oblongs.  Fold  back  the  two 
free  edges  to  meet  opposite  folded  edge. 
Open  so  that  paper  is  folded  into  halves. 


FOLDING  AND  WEAVING 


59 


Make  cuts  indicat- 
ed   from    folded    edge 
(Fig.   1).     Open  and    -- 
weave     over     3     and 
under   1. 

Fig.  2  shows  two 
res  alts  of  weaving  this 
way. 


FREE  GUTTING  AND  WEAVING 


The  child  has  dealt  with  the  relations  1,  2,  3,  4,  equal, 
■|,  ^,  ^  in  the  folding  lessons;  now  he  is  ready  to  do  some 
free  cutting.  He  should  try  to  cut  a  piece  of  paper  into  2, 
3  or  4  equal  pieces,  as  the  case  may  be. 


LESSON  I 

Material:     Paper  6''x9'\ 
Fold  and  cut  to  get  6-inch  square. 
Color  oblong,  which  is  left  with  crayola. 
Cut  into  2  equal  oblongs. 
Fold  square  into  2  equal  oblongs. 

Hold  it  so  that  folded  edge  is  down.  Now  cut  so  that 
folded  edge  is  divided  into  2  equal  parts  and  cut  is  made  half 
way  to  top  of  paper.     (Fig.  la.) 

Divide  left  half  of  folded  edge  into  3 
equal  parts  the  same  way  (&).  Divide 
right    half    same    way    (c).      Open    and 


B     A     C  weave.    Eesult   same   as   Folding  Lesson 

IIL 

60 


FKEE  CUTTING  AND  WEAVING 

LESSOX  II 


61 


Eepeat  I,  but  cut  the  oblong  which 
is  for  strips  into  4  equal  parts  instead 
of  2. 

After  folded  edge  of  mat  is  divided 
into  2  equal  parts,  divide  each  half  into 
4  equal  parts,  cutting  half  way  to  upper 
edge.     Weave. 


LESSON  III 


Material:     Two  5-inch  squares,  contrasting  colors. 

Fold  the  one  which  is  to  be  the  mat  into  2  equal  oblongs. 

It  is  time  now  to  show  the  children  how  to  cut  a  strip  for  the 

border,  along  each  end  about  ^  an  inch 

wide  and  within  half  an  inch  of  the  top 

{a,  h). 

Xow  divide  the  folded  edge  between 
a  and  h  into  3  equal  parts  and  make  cuts 
c,  d,  stopping  half  an  inch  from  the  top. 
Open. 

Cut  an  inch  strip  from  the  other  5- 
inch  square.  Teacher  cuts  first.  Child 
imitates.  This  1-inch  strip  is  discarded 
after  some  concrete  problems  have  been 
given  causing  child  to  compare  it  to  sur- 
face which  is  left.     (Relation  1  to  4.) 


62 


MATHEMATICAL  CONSTRUCTION 


Cut  this  large  surface  into  3  equal  oblongs,  making  cuts 
parallel  with  long  edges  of  surface.    Weave. 


LESSON  IV 


into  3  equal  parts 

ones,  then  other  wide  one 


Material :  Two  8-inch  squares,  con- 
trasting colors. 

Make  foundation  same  as  in  Lesson 
III. 

When  3  strips  have  been  cut,  divide 
one  of  them  into  3  equal  oblongs  cutting 
lengthwise.  Divide  middle  strip  of  mat 
Weave,  using  wide  strip  first,  then  3  small 


LESSON  V 


Material:     Two  9-inch  squares,  con- 
trasting colors. 

Foundation  same  as  Lesson  III. 

Divide    two    of   the   three    strips   for 
weaving  into  3  equal  oblongs  each. 

Divide  the  left  and  right  strips  of  the 
mat  into   3  equal  parts,  leaving  middle 
one  as  it  is. 
Weave,  using  3  narrow  strips  first,  then  the  wide  one, 
then  3  more  narrow  ones. 


h 


FEEE  CUTTING  AND  WEAVING 


63 


LESSON  VI 


B 


Material :     Two  10-inch  squares,  con- 
trasting colors. 

Fold  one  square  into  2  equal  oblongs, 

cut  the  half-inch  strips  for  border  (a,  b) 

from   folded   edge.     Divide    folded   edge 

into  2  equal  parts  at  c. 

Divide  a-c  into  3  equal  parts.     Divide  c-h  into  3  equal 

parts. 

Cut  an  inch  strip  from  other  square. 
Divide  large  surface  into  2  equal  ob- 
longs, cutting  parallel  to  long  edges. 

Cut  each  of  these  into  3  equal  parts. 
Weave  in  any  desired  way.  Over  one, 
under  one ;  over  two,  under  one ;  over  two, 
imder  two;  over  three,  under  one;  over 
three,  under  two. 


LESSON  VII 


Material:       Two     oblongs  8"x5". 
contrasting  colors. 

Fold  the  one  which  is  for  the  mat 
so  that  the  long  edges  meet.    Cut  as  in-        ^  B 

dicated  at  a,  h,  making  strips  for  border.     (Fig.  1.) 

Divide  the  fold  between  a  and  h  into  5  equal  parts,  cut- 
ting to  half  an  inch  from  top. 


64 


MATHEMATICAL  CONSTEUCTION 


Cut  an  inch  strip  from  one  of  the  long  edges  of  the  other 
oblong  from  which  strips  are  to  be  cut. 

Now  divide  this  into 
5  equal  oblongs,  cutting 
parallel  to  long  edges. 

Cut  2  of  these  ob- 
longs into  two  smaller 
equal  oblongs  each. 
Weave  thus:  one  wide 
strip,  two  narrow,  one 
wide,  two  narrow,  one 
wide.     Result:    Fig.  2, 


FREE  CUTTING  AND  WEAVING 


65 


XXXTJ-XJTJ-Xr. 


WALL  PAPER  MADE  BY  FREE-CUTTING 

Fig.  1.  Strips  any  width, 
pasted  on  background  at  a 
distance  from  one  another 
equal  to  their  own  width. 
Border,  little  oblong  strips, 
same  width  as  wall  paper 
strips,  laid  as  shown. 

Fig.  2.  Strips  any  width, 
pasted  on  background  so  that 
distance  between  strips  is 
twice  the  width  of  the  strips. 
For  border,  cut  strips  same 
width  as  other  strips,  but 
have  relation  of  length  of 
small  oblongs  to  be  cut  from 
these  strips  1  to  2.  Paste 
border  as  suggested. 


'*'?-  I 


I— I— I— I  — I 


T>  7 


^:4K4K<H<Kt 


Fig.  3. 


66 


MATHEMATICAL  CONSTRUCTION 


Fig.  3.  Eelation  of  strips  to  background  space  1  to  3. 
Border  made  from  equal  squares,  which  have  one  corner  cut 
out.     Eelation  of  this  corner  to  part  left  is  1  to  3. 


■nn 


Fl«.  4. 


Fig.  4.  Strips  grouped.  Eelation  of  narrow  to  wide 
strips  1  to  4.     Eelation  of  spaces  seen  in  background  1  to  4. 

Width  of  oblongs  in  border  twice  that  of  narrow  strips. 
Oblongs  placed  so  that  one-fourth  of  one  touches  ^  of  next  one. 

Fig.  5.  Eelation  of  strips  to  background  1  to  4.  Figures 
on  background  and  in  border  are  equal  squares  cut  on  a 
diagonal.  Strips  for  border  are  half  the  width  of  the  other 
strips. 


FEEE  CUTTING  AND  WEAVING 


67 


Fig.  5. 


PAPER  RINGS  MADE  BY  FEEE  CUTTING 


Make  curtains  or  Christmas  tree  ornaments  of  paper  rings 
cut  in  a  definite  way.  Take  an  8-inch  square  for  instance. 
Cut  it  into  2  equal  parts  free-hand.  Divide  each  one  of  these 
into  4  equal  parts.  Now  you  have  8  1-inch  strips  8  inclies 
long  to  be  divided  in  one  of  several  ^ays  according  to  the  size 
of  rings  desired. 

One  way — Bisect  them  vertically  and  horizontally. 

Another  way — ^Trisect  them  vertically  and  horizontally. 

Another  way — Quadrisect  them  vertically  and  horizon- 
tally. 

Another  way — Trisect  them  vertically  and  quadrisect 
horizontally. 

Another  way — Bisect  them  vertically  and  trisect  horizon- 
tally. 


MEASURING  AND  WEAVING 

LESSON  I 

Bank  Decoration 

(For  bank  in  Ruler  Lesson  IX,  page  93) 

Material :     A   4-ineh   square   and   an   oblong   4"x3"    of 
contrasting  color. 

Eulers:     1"2"3"4". 

Compare  oblong  to  square,  giving  problems.      ( Relation 
1  to  2.)    ■ 

Compare  rulers  and  estimate  length  and  width  of  papers 
with  them. 

Divide  the  4-inch  square  into  16  one-inch  squares,  using 
rulers  3'''2"1",  as  shown  in  regular  measuring  lessons. 

Fold  it  into  2  equal  oblongs.     Make 
cuts  indicated  on  folded  edges.     Cuts,  1, 
3,  5  are  on  the  lines.     Cuts  2,  4  are  half- 
way between. 
I   2  >  ¥  >~      '  ^'^^  ^^^®   oblong  4'''x2"   into  4  equal 

oblongs  lengthwise.     Weave. 
Make  six  of  these  and  paste  them  on  the  outside  surfaces 
of  the  bank  made  in  Euler  Lesson  IX,  page  93, 

68 


MEASURING  AND  WEAVING 


69 


LESSON  II 
Calendar  Back 

Material :  One  sheet  colored  paper 
S'^xS";  one  sheet  paper  7"x5'',  of  contrast- 
ing color,  from  which  to  cut  strips. 

Eulers:     1"2"3"4"7". 

Tell  children  length  and  width  of  paper. 
Let  them  find  out  with  their  rulers  what 
sums  equal  the  width,  5'' ;  or  the  length,  8" 
or  7' 


e 

G 

c 

-i-'- 

j    1 

1    1 

1  _!  _. 

A 

[    1 

.1     1 

'.    The}^  will  see. 
3         2         3         3 

7 

3 

3 

_F 

3         3 

2         2         13 

1 

3 

2 

4         1 

-112 

— 

1 

2 

—       — 

5       —      — 


5'  l''=5" 
4'  2"=8'' 

Now  on  long  edges  of  paper  8"x5"  with  7-inch  ruler 
make  marks  and  draw  line  a-h. 

Make  marks  c,  d  with  1-inch  ruler  and  draw  line. 

Fold  lower  edge  to  line  a-h.    Open. 

Fold  upper  edge  to  line  c-d.    Open. 

Use  4-inch  ruler  to  make  marks  g,  h,  measuring  from  left 
on  upper  and  lower  edges.  Draw  line  g-h.  Now  with  3-inch 
and  2-inch  rulers  respectively  make  the  two  inner  long-lines. 

With  1-inch  ruler  make  marks  e,  f,  and  draw  line. 

Fold  lower  to  upper  edge,  having  lines  on  the  outside. 

Cut  from  folded  edge  up  to  the  fold  near  the  top  on  every 
line  which  was  drawn.  Now  divide  each  one  of  these  free-hand 


70 


MATHEMATICAL  CONSTRUCTION 


into  two  equal  parts,  stopping  at  the  fold  near  the  top. 
mat  is  now  dividecl  into  10  half-inch  strips  and  ready 
woven. 

To  make  the  strips  divide  the 
7''x5"  paper  into  7  one-inch  strips, 
using  single  unit  rulers  to  make 
points  on  the  long  edges,  and  draw 
lines  connecting  them. 

Divide  each  of  these  hy  free-hand 
cutting  into  2  equal  parts  length- 
wise.    Weave.     Fig.  2. 

On  this  paste  a  little  art  picture 
and  a  calendar.  Braid  3  8-inch 
strands  of  raffia  together,  slip  it 
through  two  holes  punched  at  the 
top,  tie  a  knot,  and  calendar  is  ready 
to  be  hung. 


The 

to  be 


LESSOX  III 


Blotter 


Material  for  mat,  2  sheets  of  paper  7"x3",  contrasting 
colors. 

Rulers:     6''o"4"3"2"r'. 

Make  the  vertical  dotted  lines  in  Fig. 
1  with  these  rulers,  using  the  6-inch  ruler 
first  to  make  the  line  nearest  the  right; 
then  the  5-inch  ruler  to  make  the  next 
line  to  the  left  of  that  one,  etc. 


Fis-/ 


MEASURING  AND   WEAVING 


71 


Using  the  long  6-inch  ruler  first  and  then  laying  the 
5-inch  ruler  in  exactly  the  same  way,  let  the  child  see  the 
exact  difference  between  the  two.  If  the  short  ruler  were 
used  first  and  then  the  longer  one  there  would  be  an  impres- 
sion that  the  shorter  one  was  covered  up,  and  relationship 
between  the  two  could  not  be  seen.  In  building,  we  do  not 
pile  large  blocks  on  top  of  small  ones  or  the  structure  would 
be  unstable  and  fall. 

Take  these  little  rulers  from  1-inch  to  6-inches ;  pile  them 
one  on  the  other  in  regular  order  from  1  to  6,  with  the  1-inch 
underneath.  What  do  you  see?  Well,  nothing  worth  while. 
Xow  pile  them  the  other  way :  6-inches  at  the  bottom  and 
1-inch  on  top.  See  the  regular  steps  from  1  to  6.  There  is  just 
as  much  difference  between  6  and  5  as  there  is  between  5  and 
4 ;  4  and  3 ;  3  and  2 ;  2  and  1.  The  child  feels  these  relation- 
ships if  the  rulers  are  handled  in  this  way — always  making 
the  large  measurement  first  and  the  small  one  on  top  of  it. 

Fold  so  that  the  lower  touches  the  upper  edge 
and  lines  can  be  seen. 

Now  begin  to  cut  on  each  line  on  the  folded  edge. 
Now  divide  each  of  these  parts  into  4  equal  parts  and 
finish  all  cuts,  cutting  until  the  distance  from  the  top 
is  exactly  the  width  of  the  little  parts. 

Now,  on  the  other  paper  draw  the  two  vertical 
lines  in  Fig.  2,  using  the  2-inch  and  1-inch  ruler  to 

make  marks  and  6-inch 
to  draw  lines.     Cut  on 
Subdivide    each    of 


T'lf-  z 


7M>ii 

■\\lL-r,'^'=,i^ 

■vt 

ii'i 

i 

-n-a 

■ 

■ 

•1 

D 

t- 

^B. 

mm> 

■ 

I.U-L 

rj 

7V^-3 


ruler 
lines, 
these 

strips,  cutting  free-hand,  into 

4  equal  long  strips. 
Weave. 


72 


MATHEMATICAL  CONSTKUCTION 


-3E-X- 


Formula  for  weaving  Fig.   3. 

Over  3,  under  3  "]  Eepeat 

Over  3,  under  3    1 3^ 

Over  1,  under  1    1  times. 
Formula  for  weaving  Fig.  4. 

Over  4,  under  4  1  Repeat 

Over  4,  under  4    13^ 

Over  2,  under  2    \  times. 
On  top  of  this  mat  place  a  piece  of  transparent,  celluloid, 
underneath  two  or  three  sheets  of  blotting  paper.     Punch  a 
hole  in  each  corner  and  insert  brass-headed  fasteners.    This 
makes  a  pretty  blotter. 


i I i  i ;  i   It'. — L 


Ttt—  if. 


LESSON  IV 


Napkin  Eing 


Two  pieces  of  paper,  contrasting  color,  7''x2". 
Eulers:     l''2"3"4"5"6". 

Divide  one  piece  of  paper  into  2  equal  oblongs  lengthwise, 
using  the  1-inch  ruler  to  make  marks  and  the  6-inch  ruler  to 
draw  line.  Fold  so  that  short  edges  touch.  Cut  a  little  dis- 
tance from  folded  edge  on  line.  Now  divide  each  half  of  this 
folded  edge  into  3  equal  parts  by  making  cuts.  Continue  all 
these  cuts  until  the  distance  from  the  top  equals  the  width 
of  one  of  the  strips.  Open.  On  other 
paper  draw  lines  indicated  in  Fig.  1,  using 
J  6"5"4''3"2''l''  rulers  respectively  to  do  it. 
Cut  on  these  lines. 


Ti(j  .»~ 


MEASUEING  AND  WEAVING 


73 


Divide  each  one  of  these  strips  into  3  smaller  equal  ones, 
cutting  lengthwise  free-hand. 
Weave  in  any  desired  way. 


Figs.  2  and  3  show  two  different 
patterns. 

Now  put  transparent  celluloid 
on  top  and  opaque  celluloid  or  stiff 
pretty    paper    beneath.      Punch    2 


T*!^-    2. 


i 


Sew   through 


holes    at   each   end. 

these  holes  to  keep 

the  pieces  together. 

Now  roll  into  ring 

form     and     slip     a 
pretty  ribbon  or  cord  through  the  holes  and  tie  in  a  bow  or 
knot. 


TT.  *-  5 


LESSON  V 
Telephone  Pad 

Material:  One  sheet  colored  paper 
9''xl2",  one  sheet  white  paper  9"xl0". 

Eulers:     1"4"5"8". 

On  paper  9'"xl2''  draw  lines  indi- 
cated, using  rulers  8"5"4"1"  respect- 
ively. Fold  so  that  lower  edge  touches 
upper  one  and  lines  can  be  seen.  Cut 
on  lines  from  folded  edge  up  to  the  one 
which  indicates  an  inch  from  top. 

Divide  middle  one  inch  strip  into  3 
equal   parts,   cutting  free-hand   up  to 


/"       ^''  r"      s' 


cross  line  1  inch  from  top.    Open  and  press  flat. 


74 


MATHEMATICAL  CONSTEUCTION 


On  9"xlO"  paper  use  the 
l-inch  ruler  on  left  and  right 
long  edges  to  mark  them  oflE 
into  inches.  Draw  lines  with 
a  long  ruler.  Cut  on  the  lines. 
Now  we  have  10  strips  9"xl". 
Divide  5  of  them  into  3  equal 
parts  lengthwise  and  weave  as 
pattern  shows. 

If  this  mat  is  not  stiflE 
enough  paste  cardboard  under- 
neath. Punch  2  holes  in  top, 
and  hang  with  a  piece  of  braid- 
ed raffia  or  cord. 


LESSON  VI 

A  Wall-pocket  for  Letters 

Two  pieces  stiff  paper,  contrasting  colors,  5"xlO". 
Rulers:     1"2''3''7"10". 

Combinations  not  seen  by  the  children  before  in  this  book 
are  shown  in  this  lesson : 

3         3         3 

7         2         2 

—         2         3 

10—2 

7      — 

10 


MEASURING  AND  WEAVING 


75 


Aft     EC 


— 1 

1 — 

:  1 

■   ) 
*   ' 

B  H   F  D 
P>3  * 


Measuring  with  1-inch  ruler  from  upper 
and  lower  left  and  right  corners  and  drawing 
line  with  10-inch  ruler,  draw  a-h,  c-d.  (Fig.  1.) 

Measure  e-f  with  3-inch  ruler. 

Measure  g-h  with  2-inch  ruler. 

Fold  so  that  lower  touches  upper  edge  and 
lines  can  be  seen.  Cut  on  each  line  from  folded 
edge,  until  half  an  inch  from  top  is  reached. 
Divide  each  of  these  parts  except  the  middle 
one  into  two  equal  parts. 


Collect  these  rulers  and  use 
a  different  set  for  the  other 
paper.  Give  children  3"4"6'' 
8"10"  rulers. 


They  see : 

2  2"=  4" 

6 

2 

2 

2 

3  2"=  6" 

4 

8 

4 

2 

4  2"=  8" 

— 

— 

— 

— 

5  2"=10" 

10 

10 

6 

4 

2  4"=  8" 

Ff^-a 


Draw  lines  indicated  in  Fig.  2,  using  rulers  2",  4",  6", 
8".  Cut  on  these  lines.  Divide  each  one  of  the  oblongs  into 
4  equal  pieces,  cutting  lengthwise,  free-hand.     Weave. 

With  7-inch  ruler  measure  down  on  long  edges  and  fold 
on  line.    See  Fig.  3. 


76 


MATHEMATICAL  CONSTEUCTION 


Place  and  paste  ends  of  strips  under  border  after  mat 
has  been  folded  on  this  7-inch  line.  Punch  holes  and  lace  as 
indicated. 


Tf^.i 


LESSON  VII 
Circular  Woven  Basket 

Material:    An  oblong  ll"x2";  an  oblong  ll"x4",  con- 
trasting colors;  a  3|-inch  square  for  base. 

Eulers:     1"2"5"6"9"10". 

New  combinations  of  numbers  are  seen  when  paper 
ll"x2"  is  given  to  children  and  they  are  told  to  estimate 
the  length  of  it.  Xot  having  the  11-inch  ruler  in  their 
hands  there  is  no  harm  in  saying:  "This  paper  is  11  inches 
long.  Find  out  what  rulers  put  together  equal  11  inches,  the 
length  of  the  paper."  They  will  see: 
10  5  9 
16         2 


11       11       H 


MEASUEIXG  AND  WEAVING 


77 


In  addition  to  these  combinations  in  ruler  comparisons, 
they  get : 

5         9         5 
1         1         5 


rx" 


5    fc* 


g't&' 


6       10       10 

Bisect  this  oblong  (ll"x2")  lengthwise,  using  1-inch 
ruler  to  make  marks  and  10-inch  ruler  to  draw  line.  Fold 
so  that  short  edges  touch  and  line  is  visible.  Cut  on  line 
from  folded  edge  for  a  short  distance.  Xow  divide  each  part 
into  3  equal  parts.  Cut  all  lines  up  until  a  distance  from 
the  top  is  reached  equal  to  the  width  of  one  of  them. 

On  paper  ll"x4"  make  marks 
and  draw  lines  with  rulers  indi- 
cated in  Fig.  1.  Fold  so  that 
upper  edge  touches  dotted  line. 
Cut  on  fold.  Save  this  piece  for 
handle.  ISTow  cut  on  all  the  ver- 
tical lines.  Divide  every  strip  into  3  equal  parts,  cutting 
lengthwise,  free-handed.  "Weave,  letting  twice  as  much  of  the 
strips  protrude  on  one  side  as  on  the  other.  After  the  mat  is 
woven  and  pasted  into  circular  shape,  these 
smaller  protrusions  should  be  turned  in. 
They  are  the  flaps,  which  are  pasted  on  to 
the  3^"  square.  Trim  off  protruding  parts 
of  square.  The  longer  protrusions  are 
turned  outward  and  cut  into  any  desired 
decorative  form. 


T    " IT 


78 


MATHEMATICAL  CONSTRUCTION 


LESSON  VIII 


Needle-book 


B 


Material:     Two  sheets  stiff  paper  ll"x4",  contrasting 
colors;  1  sheet  thin  lining  paper  ll"x4";  one  or  2  oblongs  of 
cloth  Wx3'' ;  10  inches  of  ribbon  for  fastening  at  hinge. 
Rulers:     1"3"4"7"8"10". 

r.'i  _  ,  On  background  paper  draw 

lines  a-h,  c-d  (Fig.  1),  measur- 
ing with  1-incli  ruler  from 
upper  and  lower  edges,  or  with 
3-inch  and  1-inch  ruler  re- 
spectively from  upper  edge.  Fold  so  that  short  edges  meet 
and  lines  can  be  seen.  Cut  from  folded  edge  on  the  lines 
until  the  distance  from  the  opposite  end  is  ^  the  width  of  the 
narrow  strips.  Divide  each  of  these  strips  into  3  equal  parts, 
cutting  parallel  to  other  cuts.  Give  the  other  paper  to  the 
children.  Tell  children  length  of  the  paper  (11").  Let  them 
find  out  with  their  rulers  what  sums  equal  11  inches.  They 
will  see : 
7  8 
4        3 


3"    U-" 


7"?r" 


4" 


11         11 

In  comparing  rulers  they  will  see : 

3  3         7         3 

4  7         11 


T.lj-i 


10 


MEASURING   AXD   WEAVING. 


79 


Draw  lines  indicated  in  Fig.  2,  measuring  with  rulers 
10"8"T"4"3''l",  respectively,  from  left  edge.  Cut  on  the 
lines.  Divide  each  of  these  pieces  into  3  equal  parts,  except 
the  large  middle  one.  Divide  this  into  2  equal  parts.  Weave. 
Paste  down  all  the  edges  of  the  strips,  and  paste  lining  paper 
on  back.  Fold  so  that  short  edges  meet.  Insert  one  or  two 
pieces  flannel  (edges  pinked).  Punch  2  holes  on  fold  and 
lace  with  ribbon. 


T'  ij  3 


MEASURING 

Introduction  of 

SINGLE    UNIT    RULERS 

LESSON  I 


Fia.  I 


Scissors  Holder  (8-inch  ruler) 

Materia] :  Use  grey  school  paper,  which  is  9"xl2' 
Teacher  goes  to  wall  with  her  paper, 
lays  her  8-inch  ruler  on  it  as  indicated  in 
Fig.  1,  tlien  draws  a  line  the  whole  w'ldtli 
of  the  ruler  at  point  marked  a.  Children 
imitate  at  their  seats.  (Children  must 
keep  their  papers  in  the  same  position  on 
the  desk  until  all  ruler  work  is  done.) 

Second  step:    Teacher  lays  8-inch  ruler  on  lower  edge  of 
the  paper  and  draws  a  1-inch  line  at  6. 
Children  imitate. 

Teacher  draws  line  with  her  ruler  connecting  a  and  h. 
Children  imitate. 

Teacher  cuts  on  line.  Children  imi- 
tate. jSTow  let  children  compare  the  two 
oblongs  X  and  Y . 

How  many  sheets  of  cardboard  as  large 
Jt        as  y  can  be  cut  from  a-f      (Ans.     Two 
sheets.)      If   this    (showing  y)    costs   5c, 
80 


f/6.  & 


MEASUEING:    SINGLE  UNIT  EULEES 


81 


what  does  this  cost   (showing  x)  ?     (Ans.     Two  5c.)     If  x 
costs  a  dime,  what  part  of  a  dime  does  y  cost?     (Ans.    Half 
a   dime.)      Xow   teacher   lays   8-inch   ruler 
along  left  edge  of  large  oblong  from  upper 
left  corner  as  indicated  in  Fig.  2,     Draw  a 
line  the  width  of  the  ruler  at  c. 

Children  imitate. 

Now  lay  ruler  on  right  edge  from  upper 
right  corner  in  same  way,  drawing  line  at  d. 
Children  imitate. 

Draw  line  from  c  to  d.    Children  imitate. 

Cut  on  this  line.    Eesult :   8-inch  square. 

This  is  enough  ruler  work  for  a  first 
lesson.  Now  fold  into  two  equal  oblongs. 
Open.  Make  point  a  touch  middle  of  top 
line  at  b.     (Fig.  3.) 

Now  make  c  touch  d.     Eesult:    Fig.  4.     Cut  on  dotted 
line.     Scissors  holder  now  needs  only  to  be  pasted. 


LESSOX  II 


Mayflower  (8-inch  and  4-incli  rulers) 


Make  an  8-inch  square  as  in  previous  lesson,  but  use  white 
paper.  At  the  end  of  the  lesson  let  the  children  use  crayola 
to  color  the  hull  of  the  boat  black.  The  white  sails  will  make 
a  pretty  contrast. 

Call  the  8-inch  square  the  tablecloth  to  be  used  for  the 
Thanksgiving:  dinner. 


82 


MATHEMATICAL  CONSTRUCTION 


If  4  people  can  sit  on  one  side,  how  many  can  sit  at  2 
sides?  (Ans.  Two  4  people.)  How  many  4  sides?  (Ans. 
Four  4  people.) 

Bisect  the  square,  using  a  4-inch  ruler.  Lay  it  on  the  top 
edge  from  upper  left  corner,  drawing  line  at  the  end.    Repeat 


Beach 


on  lower  edge,  and  draw  bisecting  line. 
Fold  on  this  line. 

Now  if  the  oblong  is  the  table  and 
there  are  4  plates  on  the  short  end,  how 
many  on  the  long  one  ?  (Ans.  2,  4  or 
8  plates.)  (Some  will  say  8  plates  on 
account  of  ruler  measurement.) 

That  is  enough  ruler  work  for  this 
early  ruler  lesson. 


Fold  one  long  free  edge  to  meet  fold. 

If  small  oblong  thus  formed  is  cloth  enough  for  a  Pil- 
grim's dress,  how  many  dresses  can  she  get  from  opposite 
large  oblong?     (Ans.     3  dresses.) 

Fold  other  long  edge  to  center  fold.  Open.  Fold  the 
other  way,  making  8  small  oblongs  out  of  the  8  inch  square. 
During  process  in  giving  problems  call  surfaces  cloth  for 
aprons  instead.     Open. 

Fold  so  that  point  b  touches  center  a. 

Fold  so  that  point  c  touches  center  a. 

Compare  triangles  thus  formed  (e  and  /),  calling  them 
Pilgrims'  shawls.  If  it  took  $1.00  worth  of  yarn  to  make  e, 
how  much  did  it  take  to  make  f?  etc. 

Some  Indians  had  a  wigwam  here  (pointing  to  d).  Some 
of  them  ran  ^  mile  along  here  (pointing  to  edge  d-g-h)  to 
the  beach  to  see  the  Mayflower.    The  others  ran  along  here 


MEASUEING:    SINGLE  UNIT  EULERS 


83 


(pointing  to  edge  d-k-m)   to  the  beach.     How  far  did  the 
others  run?     (Ans.    Just  as  far,  ^  mile.) 


Fold  so  that  point  d  touches  a.  Cut  out  triangles 
Compare  size  of  triangles,  which  form  sails. 

If  large  triangle  shows  number  of 
people  who  came  over  in  Mayflower,  and 
small  one  shows  how  many  got  sick,  what 
part  of  them  got  sick? 

Many  more  questions  can  be  asked  if 
there  is  time ;  questions  about  clearing  the 
ground,  planting  corn,  chopping  down 
trees,  building  log-cabins,  etc. 


1,  2,  3,  4. 


LESSON  III 

Cup 


Material:    A  sheet  of  manila  paper  G'^xO". 
Rulers:    6''3". 
^  Pass  the  6-inch  rulers  saving,  "This  is  a  fi-inch 

ruler." 
^«»         Pass  the  3-inch  rulers  saying,  "This  is  a  3-inch 
ruler." 

Let  the  children  put  the  rulers  next  to  each  other 
(Fig.  1). 

How  many   3-inch   rulers  can   be   cut   from   the 
Ptji     6-inch  ruler? 


84  MA.THEMATICAL  CONSTEUCTION 

Show  me  the  part  of  the  6-inch  ruler  which  equals  3" ;  or, 
show  me  the  part  of  the  6-inch  ruler  which  is  just  as  much  as 
the  3-inch  ruler.     (Children  show  a.) 

Now  show  me  the  difference  between  the  6-inch  ruler  and 
the  3-inch  ruler.     (Children  show  h.) 

Show  me  the  sum  of  6"  and  3".  Children  show 
Fig.  2. 

When  in  a  later  lesson  they  use  rulers  3"6"9"', 
they  will  be  able  to  say  that  the  sum  of  3"  and  6" 
equals  9".  In  this  lesson  it  is  enough  to  have  them 
get  an  idea  of  what  the  word  "sum"  means. 

Let  them  estimate  the  length  of  the  paper  which 

they  have.     They  will  tell  you  that  it  is  the  sum  of 

6"  and  3"  long,  or  3'3''  long,  or  6"  and  i  of  6''  long. 

In  estimating  width,  they  will  say:     "The  paper 

is  6''  wide,"  or  "The  paper  is  2'3"  wide." 

Now  teacher  lays  her  6-inch  ruler  on  upper  edge 
F'^a-     and  makes  a  mark  at  a  (Fig.  3).     Children  imitate. 

Repeat  on  lower  edge,  marking  at  h. 
Draw  line  a-b.  Cut  on  line.  Compare  sur- 
faces c  and  d. 

Play  c  is  a  towel.  How  many  can  be 
made  from  d?  If  c  costs  a  nickel,  d  costs  2 
nickels.     If  d  costs  a  $1.00,  c  costs  ^  of  a  f']ff-^' 

dollar. 

Now  using  3-inch  ruler,  bisect  all  the 
edges.  Do  not  draw  the  bisecting  lines. 
They  are  not  necessary. 

Fold  this  6-inch  square  on  one  diagonal 
with  bisecting  marks  on  the  outside  (Fig.  4). 
Make  point  a  touch  mark  b. 

F«5  ^ 


MEASURING  85 

Make  point  c  touch  point  d. 

Turn    back    in    opposite    directions 
the  two  free  points  at  e.    Result:   A  drinking  cup 
for  use  on  a  train,  or  in  the  park. 


LESSON  IV 

Bookmark 

Material :    A  pretty  thin  cardboard,  a  piece  of  cord  which 
harmonizes  in  color  with  the  cardboard. 
Rulers:    2''4". 

Since  cardboard  comes  in  large  sheets  and  must  be  cut  up 
by  the  teacher,  she  may  as  well  cut  it  the  proper  dimensions, 
4"x2",  to  save  material.  But  let  the  children  have  the  two 
rulers  and  compare  them  as  in  previous  lesson,  showing  sum, 
difference,  etc.  Let  them  estimate  length  and  width  of  paper. 
Let  them  see  that  sometimes  the  paper  is  just  wide  enough 
and  does  not  need  marking. 

Lay  2-inch  ruler  on   upper    and    lower 

long  edges  respectively.    Make  marks,  but  do 

not  draw  bisecting  line.    It  is  not  necessary. 

Compare  triangle  thus  formed  to  oppo- 

^  site  surface.    If  it  is  enough  leather  for  the 

corner  of  a  book,  the  opposite  surface  is  enough  for  how 

many  corners?     If  it  costs  a  dime,  how  much  does  opposite 

surface  cost?  '  (3  dimes.) 

Fold  so  that  point  d  touches  a.  Compare  triangle  thus 
formed  to  opposite  large  triangle.     Relation  ^.     Punch  holes 


80 


MATHEMATICAL  CONSTKUCTION 


ou  the  free  edges  of  the  small  triangles  along  line  a-h.   Punch 

middle    holes    first    equidistant    from         ^ 

points  a  and  b.  Then  the  others  equi- 
distant from  center  and  points  a  and  b, 
respectively.  Lace  these  and  you  have 
a  book  mark  which  fits  on  the  corner 
of  the  page. 


LESSON  V 


Stamp  Pocket 


Material :    A  heavy,  pretty  paper. 

Eulers:    2''3"5''. 

Make  a  5-inch  square,  using  a  5-inch  ruler. 

Then  give  the  2-inch  and  3-inch  rulers  to  children.     Get 
sum  2+3=5. 

Lay  3-inch  ruler  on  upper  edge  from 
upper  left-hand  corner  and  make  mark 
a.     (Fig.  1). 

Repeat  on  lower  edge  and  make 
mark  b. 

Draw  the  line. 

Lay  2-inch  ruler  on  upper  edge  from 
upper  left-hand  corner  and  make  mark 

c.  Eepeat  on  lower  edge,  making  mark 

d.  Draw  line. 

Lay  3-inch  ruler  along  left  edge  from  upper  left  corner 
and  make  mark  e.    Eepeat  on  right  edge,  making  mark  /. 
Draw  the  line  e-f. 


D      3 


MEASUEIXG:    SINGLE  UNIT  KULEES 


87 


Lay  2-inch  ruler  along  left  edge  from  upper  left  corner 
and  make  mark  g.  Repeat  on  right  edge  and  make  mark  li. 
Draw  the  line. 

Fold   so   that   lower   edge    touches 
upper  edge.     Open.     (Fig.  2). 

Fold  so  that  left  edge  touches  line 
c-d.    Open. 

Fold  so  that  right  edge  touches  line 
a-h.     Open. 

Cut  from  e  to  fold. 

Cut  from  /  to  fold. 

Fold  so  that  point  g  touches  h. 

Fold  so  that  point  i  touches  k. 

Fold  so  that  d-h  line  touches  li-h  line. 

Fold  so  that  upper  left  corner  touches  point  I. 

Fold  so  that  upper  right  corner  touches  point  m. 

Fold  so  that  c-a  line  touches  l-m  line. 


1    u 

M   • 

I 

1    n 

K    ; 

r(9.a- 


Fold  so  that  line  a  (Fig.  3) 
touches  line  h. 

Fold  so  that  line  d  (Fig.  3) 
touches  line  c. 

Eesult:     Fig.  4. 


Fold  under  the  little  triangles  1,  2,  3, 
4  and  paste.  Result:  A  stamp  holder 
with  two  pockets.  Close  it  by  making 
lower  edge  touch  upper  one. 


88  MATHEMATICAL  CONSTRUCTION 

LESSON  VI 
Traveler's  Pocket  for  Wash  Cloth 

Material:     Oilcloth  (a  9-inch  square)  and  a  piece  of  tape 
14"  long. 

Eulers  1"  V  3"  4''. 

Handling  these  rulers  the  children  find  that: 

3  4  3  4 

13  3  1 


The  children  have  handled  a  2-inch  ruler  in  Lessons  IV 
and  V.  They  see  a  2-inch  length  in  the  making  of  this  wash- 
cloth pocket.  So,  even  though  they  haven't  the  2-inch  length 
in  their  hands  some  will  see : 


2 

4 

3 

Q 

2 

1 

2 

9 

¥ 

- 

-- 

t 

7 

7 

3 

1 

«-        :^ 

? 

<o                \o 

i 

f 

s                    i> 

-^      F 


AVith  the  1-inch 
ruler  mark  off  1  inch 
on  upper  edge  from 
upper  left  corner.  Ee- 
peat  on  lower  edge  and- 
draw  line  a-h. 


MEASUEIXG:    SINGLE  UNIT  KULEES 


89 


Lay  7-inch  ruler  on  upper  edge  from  point  a  and  make 
mark  c.    Repeat  on  lower  edge  and  make  mark  d.    Draw  line. 

Lay  3-inch  ruler  along  left  edge  from  lower  left  corner 
and  make  mark  e.  Repeat  on  right  edge  and  make  mark  /. 
Draw  line  e-f.  Lay  4-inch  ruler  on  left  edge,  beginning  at 
point  e;  mark  point  g.  Repeat  on  right  edge  and  mark  point 
h.    Draw  line  g-li. 

Cut  out  oblongs  1,  2,  3,  4. 

Compare  oblongs  5  and  6   (|). 

Compare  oblongs  6  and  8   (7). 

Compare  oblongs  1  and  8   {\). 

Compare  sum  of  5  and  6  to  7. 

Fold  over  the  flaps  8  and  9. 

Fold  7  on  top  of  them. 

Fasten  with  brass  fasteners. 


(They  equal  2-7s.) 


Cut  off  triangles  1  and  2. 
Sew  middle  of  tape  to  point  t. 
Fold  on  dotted  line  and  tie  the  tape  on 
opposite  side. 


c 

A 

;       i 

G 
E 

K 

L 

. 

M 

:           ; 

i      ; 

^ 


E 


Tf^.a. 


LESSOX  YII 

PiLGRni'S    BOXXET 

Material :    White  paper  cambric 
(a  6-inch  square). 
Rulers:    2"4"6". 
Ruler  questions : 


90 


MATHEMATICAL  CONSTEUCTION 


Put  the  2-incli  ruler  next  to  the 
4-inch  ruler. 

How  many  2-inch  rulers  can  be 
cut  from  the  4-inch  ruler? 

Show  part  of  4-inch  ruler  which 
^'s-  -•  equals  2-inch  ruler. 

Show  difference  between  4-inch  ruler  and  2-inch  ruler. 
Show  sum  of  2  inches  and  4  inches. 
Lay  6-inch  ruler  next  to  this  sum. 
How  much  is  the  sum  ? 
Teacher  writes  on  board :      2 

4 


How  many  2"  in  6'''  ? 

Show  difference  between  6"  and  4". 

In  estimating  length  of  6-inch  square,  children  will  say: 

It  is  6"  long. 

It  is  the  sum  of  4:"  and  2"  long. 

It  is  3'2"  long. 

It  is  the  sum  of  4"  and  -^  of  4"  long,  etc. 

Proceed  with  drawing  of  measurements. 
(Fig.  1.) 

Lay  4-inch  ruler  on  upper  and  lower 
edges  from  left-hand  side.  Make  marks  and 
draw  line  a-h  with  6-inch  ruler.  Use  2-inch 
ruler  in  same  way,  making  line  c-d. 

Laying  4-inch  ruler  along  right  and  left  edges  from  the 
top,  make  marks  and  draw  line  e-f. 

Laying  2-inch  ruler  in  same  way  draw  line  g-h.    Fold  top 
edge  to  line  g-li. 


MEASURING  91 

Compare  this  surface  to  opposite  one   (relation  1  to  5). 

If  this  cloth  (pointing  to  small  oblong)  costs  a  nickel, 
what  does  this  cost  (pointing  to  opposite)  ?     (Ans.  5  nickels). 

If  it  costs    a    dime?     (5    dimes.  ) 

If  it  costs  2    cents?     (5  2  cents.) 

Cut  off  this  narrow  oblong. 

Now  fold  back  ]c-l-m  on  line  g-]i.    Result :  Fig.  2. 

Make  cuts  1,  2. 

Paste  or  sew  squares  .r,  y,  z  on  top  of  one  another. 

Cut  into  two  equal  strips  the  long,  narrow  oblong,  which 
was  cut  from  the  6-incli  square.  These  are  the  ties.  Sew 
or  paste  them  on. 


LESSOR  VIII 

Stove 

Eulers:     3"6"9". 

Give  grey  school  paper  which  comes  9'"xl2".     Ask  ques- 
tions on  rulers,  bringing  out  following  number  relations: 

3         2 

3)9     3)6 
3'3"=9" 

3       6       3  6       9 

Let  the  children  estimate  length  and  width  of  paper. 
They  will  say: 
The  paper  is  4'  3''  long. 
2'  6''  long. 
9"+3"  long. 


9 

9 

6 

3 

6 

2'3"=6" 

-6 

-3 

-3 

3 

3 

3'3"=9" 

92 


MATHEMATICAL  CONSTKUCTION 


6''+3''+3"  long. 
9''+^  of  6"  long,  etc. 
3'  3"  wide. 
9"  wide. 
6"+3"  wide. 
6"+i  of  6"  wide. 
With  9-inch  ruler  mark  off  9  inches  on  upper  and  lower 
edges.      Draw   line    and    cut.      Compare    oblong   to    9-inch 
square.     Eelation  1  to  3. 

If  oblong  shows  cost  of  1  lb.  of  meat,  what  does  square 
show?     (Ans.     Cost  of  3  lbs.) 

If  it  shows  cost  of  1  pint  of  milk,  what  does  square  show  ? 
(Ans.    Cost  of  3  pints.) 

If  1  pint  costs  3c,  what  does  other  cost?     (Ans.  3'  3c 
or  9c.) 

The  children  ought  to  be  able  to  infer  9c  because  they 
found  out  that  3'3''  equals  0". 

Using  6-inch  ruler  for  mark- 
ing  and  9-inch  ruler  for  drawing 
the  line,  make  line  a-h. 

Make  marks  c-d  with  3-inch 
ruler. 

Draw  line  c-d  with  9-inch 
ruler. 

Make  marks  e-f  with  6-inch 
ruler  (laying  it  on  left  and  right 
edges  from  top). 


C                A 

K 

5                1    N 

Q 

■^^                                1 

f 

L 

M 

1 

;        1  p 
1        • 

H 


Q 


Draw  line  with  9-inch  ruler. 

Make  marks  g-h  with  3-inch  ruler. 

Draw  line  with  9-inch  ruler. 

Cut  as  indicated  (Fig.  1)  at  g,  Ji,  e,  f. 


MEASURING 


93 


Paste  h  and  m  under  /. 

Paste  n  and  p  under  o. 

Cut  out  free-liand,  cutting  so  that  the 
stove  will  have  short  legs. 

Make  oven  (a-Fig.  2),  cutting  3  sides 
of  it  free-hand.     Hinge  is  dotted  line. 

Cut  lids  from  top  of  stove.  Make  stove 
pipe  out  of  oblong,  which  was  cut  from  the 
9-inch  square. 

Give  2-inch  rulers  to  children.     (The  oblong  was  9 

Lay  2-inch  ruler  on  right  and  left  long  edges, 
marks.     Draw  line  and  cut.     Result:  Oblong  3"x2". 
3-ineh  edges  together,  making  cylinder  for  stove-pipe. 
Stove  is  made. 


x3")- 

making 

Paste 

Insert. 


LESSON  IX 

Bank 

Ptulers:     4''8"12". 

Material:    Two  12-inch  squares  of  manila  paper  cut  from 
the  school  paper  which  comes  12"xl8". 

Use  6"xl2"  pieces,  which  are 
left  in  Lesson  X.  °  ^ 

Euler  questions : 

How  many  4"  in  8"  ?  ^ 

WTiat  part  of  8"  equals  4"? 
a  of  8"). 

"What  is  the  difference  between        E 
8"  and  4"? 

Wliat  is  the  sum  of  8"  and 
4"? 


■^-^—^  I 


94  MATHEMATICAL  CONSTRUCTION 

2'4"  equals  how  many  inches? 

3'4"  equals  how  many  inches? 

What  part  of     8"   equals  4"? 

What  part  of   12"   equals   4"? 

Estimate  length  and  width  of  paper. 

It  is   12"  long. 

It  is  3 '4"  long. 

It  is  the  sum  of  8"  and  4"  long. 

It  is  the  sum  of  8"  and  ^  of  8"  long. 

Lay  8-inch  ruler  on  upper  and  lower  edges,  respectively, 
from  left  side.  Mark  a,  h  and  draw  line  a-h.  To  draw  line 
c-d  use  4-inch  ruler  in  same  manner. 

To  draw  line  e-f  lay  8-inch  ruler  on  left  and  right  edges, 
respectively,  from  top  of  paper.  Make  marks  e,  f  and  draw 
line.    To  draw  line  g-h  use  4-inch  ruler  in  same  manner. 

Fold  on  line  e-f.    Compare  the  two  surfaces  thus  formed. 

If  I  saved  $1.00  in  so  much  time  (pointing  to  small  sur- 
face), how  much  should  I  save  in  so  much  time  (pointing  to 
large  surface)  ?  If  I  save  $4.00?  (Ans.  2  $4.00,  or  $8.00— 
inference  from  ruler  measurement). 

If  a  man  earns  $8.00  in  so  much  time  (indicating  large 
surface),  how  much  would  he  earn  in  so  much  time?  (Ans. 
^  of  $8.00  or  $4.00.) 

If  a  man  earns  $8.00  and  spends  ^  of  his  money,  how 
much  does  he  spend  ? 

Make  cuts  indicated  at  d,  f,  a,  g.  Fold  and  paste  into  box 
shape. 

Next  day  make  another  just  like  it,  slipping  it  over  the 
first  like  the  cover  of  a  box.  Make  a  coin  slit  in  the  top. 
Bank  is  made.  For  decoration,  see  Measuring  and  Weaving, 
Lesson  I,  page  68. 


MEASUEING 


95 


LESSOX  X 
Envelope 

(lu  which   to   keep   little  freely  cut   units   for   decoration). 

Bulers:    5"V'7"2'\ 

Material:     Paper  left  from  previous  lesson  e^xl'^''. 

In  estimating  length  of  paper,  children  will  get  sum  of 
7"  and  5"  equals  12"  even  though  the  12'"  ruler  is  not  in 
their  hands,  for  they  have  handled  the  12 -inch  ruler  twice 
while  making  the  bank. 

In  estimating  width,  they  will  get  sum  of  5"  and  1" 
equals  6"  for  same  reason.  They  have  used  6-inch  ruler  in 
Lessons  III,  Til,  and  YIII. 

Other  numerical  values  seen: 

5         7         7         5  6'2''=12" 

2-2-5         5 

7         5         2         2 


12 

Lay  7 -inch  ruler  on  left  and  right 
edges,  respectively,  from  upper  edge ; 
make  marks  and  draw  line  a-h. 

Use  2-inch  ruler  in  the  same  way. 
Draw  line  c-d. 

With  1-inch  ruler  make  marks,  e-f-g-h. 

Fold  line  e-f. 

Fold  line  g-Ji.  (The  children  have 
no  ruler  at  their  desks  long  enough  to 
draw  this  line.) 

Cut  out  oblongs  1,  2,  3,  4. 


96 


MATHEMATICAL  CONSTEUCTION 


Fold  back  oblongs  6,  7,  and  5. 

Compare  5  to  opposite  surface.  Relation  1  to  5.  Play 
oblong  5   is  blotter. 

How  many  blotters  just  as  large  can  be  cut  from  opposite 
surface  ? 

If  large  blotting  paper  costs  a  nickel,  what  does  small  one 
cost?     (1/5  of  a  nickel  or  Ic),  etc. 

Put  paste  on  flaps  6  and  7. 

Paste  8  on  them. 

Cutting  from  c  to  e  and  from  d  to  g  makes  the  fold  over 
flap  look  better. 

LESSON  XI 

Taboeet 

Eulers:     8''5"3''1". 

Material :    An  8-inch  square  of  rather  heavy  paper. 

New  numerical  relations  seen : 

5  8  8 

3  -3  -5 


8  5  3 

Lay  5-inch  ruler  on  upper  and  lower  edges,  respectively, 
from  left  edge,  making  marks  a,  h.  o      a 

Draw  line  a-h.    Lay  3-inch  ruler  in 
same  way  and  draw  line  c-d. 

Lay  5-inch  ruler  on  left  and 
right  edges,  respectively,  from  upper 
edge ;  make  marks  e-f  and  draw  line. 

Lay  3-inch  ruler  in  same  man- 
ner and  draw  line  g-h.     Make  the 


3;  i  i         :V 


MEASUEING 


97 


n 


marks  which  are  necessar}^  to  draw  the  dotted  edges  of  the 
small  oblongs  1,  2,  3,  4,  with  a  1-inch  ruler. 
Draw  the  lines  with  the  3-inch  ruler. 

Cut  out  oblongs  1,  2,  3,  4. 

Make  cuts  indicated  at  marks  e,  f,  g,  h. 

Paste  into  box  shape.     (Fig.  2.) 

Cut  out  sides  in  any  desired  way  to  form 
legs. 


M^ 


LESSON  XII 
Chair  for  Father  Bear 
(Story  of  Three  Bears). 
Eulers:     3"6"9"12". 

Material:  School  paper  9"xl2".  Numerical  relations 
seen  in  comparisons  of  rulers  and  estimation  of  length  and 
width  of  paper. 


4 

2 

2'3"=  6" 

3"  are 

i 

of 

6" 

3)12 

6)12 

3'3''=  9" 

3"  are 

1 

3 

of 

9" 

3 

2 

4'3"=12" 

3"  are 

1 
4 

of 

12" 

3)    9 

3)   6 

2'6"=12" 

3       3       6       6       3       6       9       9     12     12     12 
6       9       6       3       3     -3     -3     -G     -9     -fi     -3 


9     12     12       3       3       3       G 
12       9 


6       9 


98 


MATHEMATICAL  CONSTRUCTION 


1 J r— — - 

7  :        i    H^ 
?   :  9    :       ■; 

/    '    2         3        5' 

I       I       I 


Lay  9-irich  ruler  on  upper  and 
lower  long  edges  respectively,  from 
left  edge.  Make  marks  a,  b.  Draw 
line  a-h.  Lay  6-inch  ruler  in  same 
way  and  draw  line  c-d.  Lay  3- 
inch  ruler  same  way  and  draw  line 
e-f.  "      "      ^ 

Lay  6-inch  ruler  on  left  and  right  short  edges,  respec- 
tively, make  marks  g-h  and  draw  line  g-h. 

Lay  3-inch  ruler  in  same  way  and  draw  line  i-j. 

Make  cut  at  h. 

Fold  back  surface  1-2-3.     Compare  to  opposite  surface. 
(Relation  1  to  3.) 

If  it  takes  a  roll  of  wall  paper  for  this  wall  (indicating 
small  surface),  how  many  for  this?     (3  rolls.) 

If  it  takes  3  rolls  for  small  wall,  how  many  for  large 
wall  (3'3  rolls  or  9  rolls-inference  from  rulers). 

Make  cuts  indicated  at  d  and  a. 

Fold  back  square  4. 

Now  compare  surface  1-3  to  opposite  surface.     Relation 
1  to  4. 

If  wall  paper  for  small  surface  costs  $3.00,  paper  for  large 


HJl 


wall  costs  how  much?  (4'$3.00  or  $12.00- 
inference  from  ruler  measurement.) 

If  it  took  12  pails  of  plaster  for  large  wall, 
how  many  for  small  wall?  {\  of  12  pails  or  3 
pails),  etc. 

Make  cut  indicated  at  i. 

Fold  and  paste  so  that  square  5  is  under  3 ; 
4  under  6 ;  and  7  under  3. 


MEASURING:    SINGLE  UNIT  EULERS 


99 


Fold  and  paste  surface  1-2  under  8-9. 

Cut  out  sides  to  form  legs  in  any  desired   design. 

LESSON  XIII 
Mother  Bear's  Chair 

Rulers:     2"4"6"8". 
Material :    StifE  paper,  G"x8'\ 

In   comparing  rulers   and   estimating  length   and   width 
of  paper,  these  numerical  relations  are  seen : 


2'2"=4''  2"  are  ^  of  4" 

3'2"=6"  2"  are  i  of  6" 

4'2''=8"  2"  are  ^  of  8" 

2'4''=8"  4"  are  ^  of  8" 

2       2  2       4       8       8 


4  2  3 

2)T       4)T       2)T 

2 
2)4 


6       6       4 


2 


6 


-6-4-2-4 


-2 


468824624       2 
Use  rules  2"'4"6"  to  divide  surface  into  12  squares,  as  the 
g//g//g//  p^igi-g  yxQ-re  nscd  in  previous  lessons. 

Make  cuts  indicated  at  a  and  h. 
Fold  back  surface  1-2.     Compare 
S     it   to   opposite   surface.      Relation    1 
to  5. 

Make  cut  indicated  at  c. 
Fold  back  surface  3-4. 
Compare  this  to  opposite  surface. 
Relation  1  to  4. 


■ 
3 

' 1 

i 

!  ^ 

? 

to 

i  2. 

7 

f 

f 

1 

100  MATHEMATICAL  CONSTEUCTION 

If  this  floor  (showing  surface  3-4)  requires  2  square  yards 
of  carpet  to  cover  it;  what  does  this  floor  require?  (showing 
large  surface.)  (Ans.  4'2  square  yards  or  8  square  yards- 
inference  from  rulers). 

Make  cut  indicated  at  d. 

Fold  and  paste  so  that  square  I  is  under  5 ;  6  is  under  2 ; 
3  is  under  5. 

Fold  back  and.  paste  7-8  on  9-10  to  strengthen  back  of 
chair. 

Make  legs  and  back  same  design  as  Father  Bear's  Chair. 


LESSON  XIV 
Baby  Bear's  Chair 


Eulers :     l''2"3"4". 

* ^ i       Material:     A  4-incli   square   of   stiff  paper. 

Divide  this  into  16  1-inch  squares  by  using  rulers  V2"2>". 

Cut  off  one  row  of  squares. 

Play  both  surfaces  are  rugs  with  fringe  on  short  edges. 

(Fig.  1.) 

If  1  yd.  of  fringe  is  on  one  short  edge  of  Tug  B,  how 
many  yds.  are  on  one  short  edge  of  rug  A  ? 

How  many  yds.  on  both  short  ends  of  af  (2''3  yds.,  or 
6  yds.-inference  from  former  ruler  measurement.) 

How  many  yds.  on  both  short  ends  of  rug  5?     (2  yds.) 

How  many  times  as  much  fringe  must  we  have  for  the 
big  rug?     (3  times  as  much.) 

3'2  yds.  are  how  many  yards?  (Inference  from  ruler 
measurement  in  previous  lessons.) 


MEASURING:    SINGLE  UNIT  RULERS 


101 


Make  cuts  indicated  at  a,  h,  c,  d. 

Fold  and  paste  chair  as  in  previous 
lessons.  Make  legs  and  back  same  design 
as  father  and  mother  bears'  chairs. 

Number  relations  seen: 


2 

9'-|  /'_9// 

V  is  \  of  2" 

1 

1 

1 

2 

2)1 

3'1"=3'' 

V  is  \  of  2," 

1 

2 

3 

2 

4'1"=4" 

\"  is  \  of  V 

2 

3 

4 

4 

2'2''=^" 

2"  are  \  of  ^" 

4 
-3 

4 
-2 

4 
-1 

LESSON  XV 

Father  Bear's  Bed— Rulers   :3''  6"  9"  12". 

Material :    2  sheets  stiff  paper,  9'''xl2". 

Divide  paper  into  12  small  squares,  using 
rulers  3'"6"9",  as  in  Lesson  XIL 

Make  cuts  indicated  at  a,  &,  Fold  back 
squares  1-2.  Compare  this  surface  to  opposite 
large  surface.  (Eelation  1  to  5).  If  it  is 
material  for  1  sheet,  how  many  sheets  can 
be  made  from  large  surface  (5  sheets).  Fold 
back  3.    Make  cut  indicated  at  c.     Fold  back  4,  5,  6 


A B 

V        '        :i     3 

I 

t>    1  •  ^ 


Com- 


102 


MATHEMATICAL  CONSTRUCTION 


pare  4,  5,  6  to  opposite  surface.  Eelation  1  to  2.  If  this  is 
material  for  a  pillow-case,  how  many  can  be  made  from  large 
surface?     (Ans.    2  pillow-cases.) 

If  it  takes  3  minutes  to  hem  one  pillow-case,  how  long 
will  it  take  to  hem  2?  (Ans.  2'3  minutes,  or  6  minutes.) 
Make  cut  d.    Fold  and  paste  into  box  shape. 

This  is  enough  for  one  lesson.  Next  day  take  the  other 
9"xl2"  sheet  of  paper  and  make  another  box  shape  exactly 
the  same.  Compare  volumes  of  these  two.  If  one  holds 
3  lbs.  of  feathers,  the  other  holds  3  lbs.  of  feathers. 

Now  cut  one  box  shape  so  that  it  will  hold  half  the 
amount  of  feathers.     Eye  measurement  only. 

Xow  if  small  box  holds  3  lbs.  of  feathers,  large  one  holds 
6  lbs.  If  large  one  holds  12  lbs.,  small  one  holds  ^  of  12  lbs., 
or  6  lbs. 


F.j.a.. 


From  small  box  shape  cut 

so  that  the  legs  of  the  bed  are 

formed. 

From  large  box  shape  cut 

half-way   down   on   the   edges 

a  and  h.    Cut  across  on  line  c. 

This  is  the  foot  of  the  bed. 

Leave   opposite   end   as   it  is. 

This  is  the  head  of  the  bed. 
Now  to  get  the  sides,  cut  down  half-way  again  on  what  is  left 
of  edges  a  and  h.    Then  cut  across  to  head  of  bed. 

Paste  Fig.  3  on  Fig.  2  and  bed 
is  constructed.  Numerical  rela- 
tions found  in  Lesson  XII  are 
reviewed. 


MEASUETXG:    SINGLE  UNIT  RULERS  103 

LESSON  XVI 

Mother  Bear's  Bed— Rulers  :  2"4"6"8". 

Materials :    2  sheets  stiff  paper  G'^xS", 
Make   just   as   father's    bed   was   made,   but   use    rulers 
2"4"G"8"  instead.     Numerical  relatious   found  in  Lesson 

XIII,  reviewed. 

LESSON  XA^I 

Baby  Bear's  Bed— Rulers:  1"2"3"4". 

Material :    2  sheets  stiff  paper  3'''x4". 
Make  the  same  as  Father's  and  Mother's  Bed,  but  use 
rulers    1"2"3"4".      Numerical    relations    found    in    Lesson 

XIV,  reviewed. 

LESSON  XVIII 

Bowl  for  Father  Bear— Rulers:  12"4"3"1". 

Material :    Piece  of  paper  12"x4"  and  a  4-iuch  square. 

Numerical  relations  seen : 
4  3 

3)~12"  4)12"  3'4"=12'' 

4'3"=12" 
3         4         4 
1       -1       -3 


1 


104  MATHEMATICAL  CONSTRUCTION 


On  left  and  right  edges, 
measuring  from  upper  edge 

}iLmLilJiiJlii<;  _  .c.  -  ■Uii'jt'iunila        with     3-ineh     ruler,    make 

marks  a,  h.    Draw  line  a-b. 
Fold  so  that  lower  edge  of 
paper  touches  line  a-h. 

Cut  on  the  fold.  Compare  this  long  narrow  oblong  to 
large  surface.  Eelation  1  to  7.  Now  fold  on  line  a-h.  Then 
on  the  long  narrow  oblong  c,  make  vertical  cuts  about  ^  of 
an  inch  apart  up  to  the  line  a-h. 

Paste  right  edge  on  left  edge  to*  form  c^dinder,  seeing  that 
the  little  cuts  on  the  bottom  turn  inward.  These  are  the  flaps. 
Put  paste  on  the  bottom  of  the  flaps.  Lay  this  cylinder  on 
the  four-inch  square  with  flaps  down.  When  the  flaps  are 
pasted  to  the  square  cut  off  the  parts  of  the  square  which 
protrude.    Bowl  3  inches  high  is  made. 


LESSON  XIX 

Mother  Bear's  Bowl— Rulers:  12"3''3"1". 

f 

Material:     Piece  of  paper  12"x3";  a  4-inch  square. 

Number  relations  seen  in  this  lesson  : 

4 

4'  S'^^lg"  ^ 

12'1"=12''  2)12"        ~       ~T       ~ 


MEASUKING:    SINGLE  UNIT  KULEES  105 

Lay  2-inch  ruler  on  left 
and  right  edges  respectively. 
Make  marks  a,  b.    Draw  line 


that  lower  edge  touches  line 
a-h.  Cut  on  fold.  Compare  this  to  large  surface.  Relation 
1  to  5.  Fold  on  line  a-b.  Compare  this  long  narrow  oblong 
to  opposite  surface.  Relation  1  to  4.  If  it  is  a  board  in  the 
floor  what  is  the  other?  (4  boards). 

If  it  costs  3c,  what  does  other  cost?     4'3c  or  12c. 

Make  the  little  cuts  for  flaps  up  to  line  a-b. 

Paste  right  on  left  edge,  seeing  that  flaps  turn  inward. 
Put  paste  on  bottom  of  flaps.  Set  cylinder  so  that  flaps  have 
their  paste  side  down  on  the  4-inch  square.  Cut  away  pro- 
truding surface  of  square.  ]\Iother  Bear's  Bowl  2"  high  is 
made. 

LESSOR  XX 
Baby  Bear's  Bowl— Eulers:  12"2"1". 

Material:     Strip  of  paper  12"x2";  a  4-inch  square. 

On  left  and  right  edges 
measuring  from  upper  edge 
make  marks  a,  b  with  1-inch 
ruler.  Draw  line  a-b  with 
12-inch  ruler.  Fold  so  that 
bottom     edge     touches     line 

a-b.  Cut  on  fold.  Compare  the  oblong  cut  off  to  large 
surface.  Relation  1  to  3.  Fold  on  line  a-b.  Make  vertical 
cuts  up  to  line  a-b  on  oblong  c  for  flaps.  Paste  right  edge 
on  left  edge,  turning  flaps  inward.     Put  paste  on  bottom  of 


JWiLii'JIj'L'iJi .   _c  -   -   u.i>iiiwmv 


e 


lOG  MATHEMATICAL  CONSTEUCTION 

them,  set  on  4-inch  square  and  cut  off  protruding  part  of 
square.     Baby  Bear's  Bowl  is  V  high. 

If  baby's  bowl   contains  a 

pint   of   porridge,   what   does 

papa's     contain      (3     pints)  ? 

What   does   mamma's   contain 

(2  pints)  ?     The  child  cannot 

say  1  quart,  unless  you  have 

had  the  actual  quart  and  pint  measurements  in  your  room 

and  he  has  measured  and  found  out  that  a  quart  equals  two 

pints. 

The  child  should  deal  with  the  actual  measures  first. 
Then  after  he  has  seen  the  relations,  let  him  draw  infer- 
ences, by  applying  those  relations  in  other  ways. 

For  instance:  When  we  made  a  bowl  for  the  baby  bear, 
and  another  with  twice  its  capacity  for  the  mother  bear,  we  did 
not  make  them  with  the  actual  capacity  of  a  pint  and  quart. 
But  we  may  assume  that  the  baby's  holds  a  pint.  Then  the 
child  infers  that  the  mother's  (having  twice  the  capacity) 
holds  a  quart,  because  he  has  previously  discovered  through 
actual  measurement  that  a  quart  is  twice  as  much  as  a  pint. 


LESSON  XXI 

Pail  for  Jack— Eulers  :  9"5"4"1". 

Material:     Pretty  colored  paper  9"x5'';  a  3-inch  square. 
Number  relations  seen: 


MEASURING:    SINGLE  UNIT  RULERS 


107 


4       9 
-i     -5 


0 

-4 


9     5     1 


9 
On  left  and  right  edges,  meas- 
uring from  upper  edge  with  4-inch 
ruler  make  marks  a,  h.  Draw  line 
a-b.  Fold  so  that  lower  edge 
touches  line  a-h.  Cut  on  fold. 
Now  fold  on  line  a-h.  Make  little 
vertical  cuts  on  oblong  c  free-hand  about  ^  of  an  inch  apart 
up  to  line  a-h.  Paste  right  edge  on  left  one,  flaps  turning 
in.  Put  paste  on  bottom  of  flaps,  lay  on  3-inch  square  and 
cut  away  part  of  square  which  is  left  on  the  outside.  Use 
first  strip  x  which  was  cut  off  as  a  handle  for  Jack's  pail. 


LESSON  XXII 

Jill's  Pail— Rulers  :  9"3"2"1". 

Material:     Prettv  colored  paper  9"x3";  a  3-inch  square. 

Lay  2-inch  ruler  on  left  and 
right  edges  respectively  from  up- 
per edge. 

Make  marks  a,  h  and  draw 
line  a-h.  Fold  so  that  lower  edge  touches  line  a-h.  Cut  on 
fold.  Use  this  strip  x  for  handle.  Fold  on  line  a-h.  Cut  c 
into  flaps.  Paste  right  edge  on  left  one,  turning  flaps  in. 
Put  paste  on  bottom  of  flaps  and  paste  the  cylinder  on  to  the 


^[liliJlijiiiii 


-i— .JUliiJJUl/ 


108 


MATHEMATICAL  CONSTEUCTION 


A 


3-inch  square.  Cut  off  protruding  parts  of  square.  Paste 
on  the  handle.  Jack's  pail  is  4"  high. 
Jill's  pail  is  2"  high.  Compare  vol- 
umes. If  Jill's  holds  a  quart  of  water, 
Jack's  holds  2  quarts. 

If  Jack's  holds  a  gallon  of  water, 
Jill's  holds  1  of  a  gallon.  If  Jill's 
holds  a  pint  of  water,  Jack's  holds  2 
pints  or  1  quart  (if  children  have  had 

pint  and  quart  measurements). 

If  a  pint  costs  3c,  what  does  a  quart  cost?     (2'  3c  or 

6c)  etc. 


J. 


LESSOX  XXIII 


Fox's  Dish— EuLERs:  5''2"1' 


Material :     Paper  5"x 


^A 


s" 


MMWsjmm 


when  it  is  finished. 
Fold  on  line  a-b. 


I" ;  di  2-inch  square. 

Using  1-inch  ruler  make  marks 
a,  h.  Draw  line  a-b  with  5-inch 
ruler.  Fold  so  that  lower  edge 
touches  line  a-b. 

Cut  on  fold.  Use  pieces  of  this 
strip  to  make  side  handles  for  dish 


Cut  c  into  flaps.     Paste  right  on  left 
edges,  flaps  turning  in.     Put  paste  on  bottom  of  flaps.    Lay 


MEASUEING:    SINGLE  UNIT  EULEBS  109 

cylinder  flaps  downward  on  2-inch  square.     Cut  off  parts  of 
square  which  project  beyond  cylinder. 

LESSON  XXIV 

Stork's  Dish— Rulers:  5"7"6"1". 

Material:     Paper  5"x7'';  a  2-inch  square. 
Number  relations  seen: 
s-  5     6       6       7       6       7 

11-1-1     -5     -6 


<    ? 


6     7       5       6       11 

Lay  6-inch  ruler  on  left  and  right  edges 

respectively.    Make  marks  a,  b.     Draw  line 

a-b.     Fold  so  that  lower  edge  touches  line 

a-b.    Proceed  as  in  previous  lesson  to  make 

the  Stork's  dish.     Use  strip  cut  off  for  side  handles, 

Relation  of  Fox's  dish  to  Stork's  dish 

is  1  to  6.    If  Fox's  dish  contains  Ic  worth 

of  meat,  what  does  Stork's  contain?  (6c 

worth, ) 

If  Fox's  dish  contains  2c  worth,  what         ^r=¥? 
does    Stork's    contain?      (6'    2c   worth    or  ^- — J 

12c  worth.) 

LESSON  XXV 

Handbag— EuLERS :  9''6"3"2"1". 

Material :     Paper  9"6'' ;  also  a  strip  9"xl''. 
Number  relations  reviewed — a  great  many  seen,  in  com- 
paring rulers  and  estimating  length  and  width  of  paper. 


110 


MATHEMATICAL  CONSTRUCTION 


6 

2 

1 

3 
2 

2'3": 

3'2": 

=6" 
=6" 

c-  c 

E   3 

3 

■^i    : 

- 

- 

3'3'^ 

=9" 

/4 

.     ;  / 

9 

3 

1 

^i  i 

i      1 

6 

w  o 

F  J 

9+3=6+6 


Lay   3-inch  ruler  on   left   and   right 
^.^^  edges    respectively.     Make    marks    a,    h; 

I  (\\  draw  line  a-h  with  9-inch  ruler. 

— ' -^*  Lay  2-inch  ruler  on  upper  and  lower 

edges,  respectively,  measuring  from  left 
edge  make  marks  c,  d,  and  draw  the  line. 
Lay  2-inch  ruler  on  upper  and  lower 
edges  respectively,  measuring  from  right 
edge;  make  marks  e,  f  and  draw  line. 

Lay  1-inch  ruler  on  upper  and  lower  edges  respectively, 
measuring  from  left  side;  make  marks  g,  h,  and  then  draw 
line  g-li. 

Doing  same  on  right  side  make  marks  i,  j,  and  then  draw 
line  i-j. 

Make  cuts  indicated  at  a  and  h.  Fold  on  lines  c-d  and 
e-f. 

Xow  fold  back  on  lines  g-h  and  i-j.     Fold  on  line  a-h, 
so  that  all  these  small  folds  are  inside. 
Paste  oblongs  1  and  2  together. 
Paste  oblongs  3  and  4  together. 

Cut  the  9"xl"  strips  into  two  equal  narrow  strips  for 
handles — eye  judgment  only. 


MEASURING:    SINGLE  UNIT  RULERS 


111 


LESSON  XXVI 
Pencil-box  with  Lid — Rulers:  10"3"3"o". 

Material:     Thin,  smooth  cardboard  10"x6". 
Number  relations  seen : 

2  5       3         2'3"=  6" 

3  5       3         3'2"=  6" 
-     _  5'2"=10'' 


5     10       2 
2 

10 


>^ 

Y 

- 

— 

Lay  5-inch  ruler  on  left  and  right  edges,  measuring  from 
upper  edges ;  make  marks  a,  h  and  draw  the  line. 

Lay  3-inch  ruler  in  same  manner;  make  marks  c,  d,  and 
draw  the  line. 

Lay  2-inch  ruler  in  same  manner;  make  marks  e,  f,  and 
draw  the  line. 

Make  marks  g,  Ti,  with  1-inch  ruler;  draw  line  with  long 
ruler. 

Make  marks  i,  j  with  1-inch  ruler;  draw  line  with  long 
ruler. 

Cut  out  oblongs  x  and  y. 

Make  cuts  indicated  at  a,  h,  c,  d. 

Fold  up  and  paste. 

Surfaces  can  be  compared  by  giving  concrete  problems 
as  in  former  lessons. 


112 


MATHEMATICAL  CONSTEUCTION 


LESSON   XXVII 

Match-safe 

Colored  smooth,  thin  cardboard  10''x4"  for  back. 
Colored  smooth,  thin  cardboard  6"x5"  for  box. 
Sandpaper,  3''x2". 
Eulers:     5"4:"3"2''V\ 

Let  children  estimate  10''  length  of  cardboard  with  these 
rulers.     They  will  say : 

5       5       4       3       4       3       2 

4       3       3       5       4       3       2 

12       2       12       3       2 

10     10       1       1     10       1       2 
10     10  10       2 


10 


On  the  cardboard  which  is  6"x5",  lay 
5-inch  ruler  on  upper  and  lower  edges, 
respectively,  measuring  from  left  edge ; 
make  marks  a,  h,  and  draw  line.  Lay 
4-inch  ruler  in  same  way;  make  marks 
c,  d,  and  draw  line.  Lay  2-inch  rule 
in  same  way;  make  marks  e,  f  and  draw  line.  Lay  1-inch 
ruler  in  same  way ;  make  marks  g,  h  and  draw  line. 


H  P 


6  6 


MEASUEING:    SINGLE  UNIT  EULERS 


113 


^a 


o 


Lay  3-inch  ruler  on  left  and  right  edges, 
respectively,  measuring  from  upper  edge;  make 
marks  i,  j  and  draw  line. 

Lay  2-inch  ruler  in  same  way;  make  marks 
h,  I  and  draw  line. 

Cut  out  oblongs  1,  2,  3,  4.  Make  cuts  indi- 
cated at  i,  j,  h,  I.  Fold  and  paste  into  box 
shape. 

Paste  box  and  sandpaper  on  large  card- 
board— spacing  agreeably. 

LESSOR  XXYIII 


Wood-box— Rulers  :  3"6"9"12"— 2"4"6"8". 

3"      ^"     6."     T-  Material:     Stiff     paper     8"xl2". 

Using  2"4"6"8"  rulers  to  divide  the 
short  edges  into  4  equal  parts,  gives  a 
good  review  of  number  relations  found 
in  Lessons  XIII  and  XVI. 

Using  3"6''9"12"  rulers  to  divide 
long  edges  into  4  equal  parts  gives  a 
review  of  number  relations  found  in 
Lessons  XII  and  XV.  Make  cuts  in- 
dicated in  Fig.  1. 


9'    : 


Fold  and  paste  into  box  shape.  (Fig.  2.) 
Cut  on  dotted  lines  a  and  h.  In  comparing 
surfaces  and  lines,  speak  of  cost  of  coal  or 
wood.  Use  the  words  "ton  of  coal,"  "cord  of 
wood."  They  hear  measurement  words  used  in  connection 
with  things  which  require  those  measures. 


^ 

, 1 

•p. 

B, 

y 

114 


MATHEMATICAL  CONSTRUCTION 


LESSON  XXIX 
Sled— Rulers  :  5"4"3''2''l'^ 


Material:     Paper  5"x3". 
Number  combinations  seen : 


1" 

«- 

1" 

B 

A 

2  3     4     3 
1112 

3  4     5     5 


3  2 
1  2 
1     1 


5     5     etc. 

Make  vertical  dotted  lines  with  4-inch  and  1-inch  rulers, 
respectively,  measuring  from  left  edge.  Make  horizontal 
dotted  lines  with  2-inch  and  1-inch  rulers,  respectively,  meas- 
uring from  the  top. 

Cut  out  rectangles  a  and  6. 

By  folding  back  rectangles  formed  by  dotted  lines  there 
are  surfaces  to  compare  which  show  the  relation  1  to  2; 
1  to  3 ;  1  to  4. 

Play  these  rectangles  are  sidewalks  covered  with  snow  to 

be  shoveled.     If  a  boy  can  shovel  this    (pointing  to  small 

-surface)    in   1  hour,  how  many  hours  will  it  take  him  to 

shovel  this   (pointing  to  the  opposite  surface).     Result  de- 


MEASUEING:    SINGLE  UNIT  EULEES 


115 


pends  on  the  surfaces  wliich  the  teacher  is  comparing,  of 
course.     Suppose  relation  to  be  1  to  2.     (Ans.  2  hours.) 

If  he  shovels  the  small  walk  in  2  hours,  he  can  shovel 
the  large  one 

in  2x2  hours,  or  4  hours. 
in  2x3  hours,  or  6  hours. 
in  2x4  hours,  or  8  hours. 
in  2x5  hours,  or  10  hours. 
in  2x6  hours,  or  12  hours. 

Many  children  will  be  able  to  give  those  results  on  account 
of  former  experiences  with  single  unit  rulers. 


LESSON^  XXX 


Pushcart— EuLERS :  7"5"4"1''2''. 


Materials:     Stiff  school  paper  9"x6". 

Ask  children  how  long  and  wide 
tlie  paper  is  without  ^^sing  any  ruler. 
They  have  handled  the  9-inch  and 
G-inch  rulers  so  often  that  they  ought 
to  know  how  they  look  by  this  time. 
Now  tell  them  to  measure  the 
length  with  the  rulers  which  they 
liave.  They  learn  that  the  sum  of 
the  7-inch  and  2-inch  rulers  equal  9";  that  the  sum  of  the 
5-inch  and  4-inch  equal  9". 

In  measuring  the  width  they  learn  that  5"  and  1"  equals 
6";  that  4"  and  2"  equals  6". 


116 


MATHEMATICAL  CONSTEUCTION 


TfiiZ 


Cut  on  dotted  line  drawn  between  marks  made  on  upper 
and  lower  long  edges  with  7-inch  ruler. 

The  rectangle  a  is  now  2"x6".  With  the  2-inch  and 
4-inch  rulers  divide  this  into  3  2-inch  squares. 

Cutting  off   the   corners   of  two   of  these  squares   in   a 
curved  line  gives  two  circles  to  be  uesd  as  wheels. 
Cut  remaining  square  into  two  equal  pieces  to  be 
used  as  supports  for  the  axles,  which  are  tooth- 
picks. 

Now  cut  on  dotted  line,  which  is  drawn  be- 
tween  marks   made    on   right   and   left   edges   with   5-inch 
ruler. 

Cut  rectangle  h  (Fig.  1)  into  two  equal  narrow  strips 
to  be  used  as  shafts.  From  c  (7"x5")  the  box  shape  is 
made.     (Fig.  3.) 

With  1-inch  ruler  make  marks  a,  h 
on  upper  and  lower  edges,  measuring 
from  left  side.  Draw  line  a-h.  With 
1-inch  ruler  make  marks  c,  d  on  upper 
and  lower  edges,  measuring  from  right 
side.    Draw  line  c-d. 

With  4-inch  ruler,  measuring  from 
top  on  the  left  and  right  edges,  make  marks  and  draw  line  e-f. 
With  1-inch  ruler,  measuring  from  top,  make  marks  on 
left  and  right  edges  and  draw  line  g-h. 
Make  cuts  indicated  at  e,  f,  g,  h. 
Fold  and  paste  into  box  shape. 

Paste  the  little  supports  for  the  axles  on  the  under  side 
of  the  box  after  the  axles  have  been  laid  in  place.  Put  on 
the  wheels  and  paste  on  the  shafts.     (Fig.  4.) 


r/^  3 


MEASURING:    SINGLE  UNIT  KULEES 


117 


In  comparing  lines  and  surfaces  in 
this  lesson,  base  concrete  problems  on 
roads,  trees,  houses,  stores  or  parks; 
things  which  the  banana-man  sees  when 
he  pushes  his  cart  through  the  streets. 

Ask  about  time  or  number  of  men  required  or  material 
needed  to  pave  or  sprinkle  roads;  to  cut  or  sprinkle  grass 
in  the  park;  to  plant  trees  in  a  row.  Ask  about  amount  of 
houses  of  equal  size  which  can  be  built  on  compared  surfaces. 


LESSON  XXXI 

GOCART— EULERS  :  2"4"6"8". 
Material:     2  sheets  of  stiff  paper,  one  10"x8" ;  the  other 

In  estimating  length  of  paper  new  number  combinations 
seen  are : 
8       6 
2       4 


10     10 


Some  children  ought  to  know 
that  the  paper  is  10"  long,  for  they 
talked  about  10"  in  Lessons  XXVI 
and  XXVII. 

Measuring  on  upper  and  lower 
long  edges  from  left  side  with  the 
8-inch    ruler    on    the    large    paper, 
make  marks  and  draw  line  a-b. 
^^  Cut  on  dotted  line  a-h.     Com- 

'^  pare  c  to  dg,  with  questions  about 

material  and  cost  of  material  for  doll  quilts  and  mattresses. 


, A__ 

I 

D       : 

;     C 


118  MATHEMATICAL  CONSTRUCTION 

From  c  measure  and  cut  4'3"  squares,  using  the  6'''4"2'' 
rulers.  From  these  squares  cut  free-hand  the  four  wheels, 
keeping  them  3"  in  diameter. 

Measuring  on  left  and  right  edges  ^ 

of  8''  square  from  the  upper  edge  with  '        ■* 

the  6-inch  ruler  make  marks,  and  draw         *  j^ 

line  e-f.     Cut  on  line  e-f.     From  g,  U 

four  equal   long  narrow  strips   are   to         '               — 
be  cut.     Two  of  these,  to  be  used  for 


the  handle,  need  a  2-inch  mark  at  one  p,^  ^ 

end  of  each. 

Fold  so  that  line  a  (Fig.  2)  touches  this  mark. 
Place    short-folded   parts   of   these   strips   together    and 
paste,  forming  handle.     (Fig.  2.) 

Mark  other  two  strips  as  indicated  in  Fig.  3. 

Fold  on  marks.     Paste 

fl    on    &.     These    are    the 

springs  to  be  pasted  under 

_.  j-^  the    box-shape    when    it    is 

made,     putting    toothpicks 

through  the  inch  square  sides  of  the  springs  for  axles. 

To  make  box-shape  use  d.  (Fig. 
1.)     It  is  8''x6''.  c  A 

Measuring  with  6-inch  ruler  on 
upper  and  lower  long  edges  from  left 
edge  make  marks  and  draw  line  a-h. 
In  same  way  using  2-inch  ruler  draw 
line     c-d.     Measuring     with     4-inch 


EZT 


ruler,  on  left  and  right  edges  from  ^'3  h- 

upper  edge,  make  marks  and  draw 

line  e-f.    In  same  way  using  2-ineh  ruler  draw  line  g-h. 


MEASUEING:    SINGLE  UNIT  RULEES 


119 


Make  cuts  indicated  at  a,  b,  c,  d.  Fold  and  paste  into 
box-shape. 

This  is  enough  for  one  lesson.  Next  day  give  other  piece 
of  paper  8"x6"  to  children.  Make  another  box  shape  from 
it  like  Fig.  4.  Compare  volumes  of  these  two  which  are 
equal. 

When  mamma  goes  marketing  she 
takes  baby  with  her  in  the  go-cart.  If 
one  volume  holds  a  peck  of  potatoes,  the 
other  holds  a  peck  of  potatoes. 

If  one  holds  half  a  peck  of  apples 

the  other  holds  half  a  peck  of  apples. 

If  one  holds  5  pounds  of  sugar,  the  other 

holds  5  pounds,  etc. 

Now  stand  last  box  up  on  end  and  inside  of  the  first 

box  which  has  the  springs  and  axles  under  it.     Put  on  the 

wheels  and  paste  on  the  handle. 


LESSON  XXXII 
Cradle— Rulers  :  1"9"5"4". 

Materials:     2  sheets  of  stiff  paper  10"xl6".     1  sheet  of 
stiff  paper  12"  square. 

Use  one  sheet  of  the  paper,  10"x6",  first. 

In  estimating  length,  get  new  relations: 
9       5 
1       4 

10       1 


10 


120 


MATHEMATICAL  CONSTKUCTION 


T'l^l 


Measure  on  upper  and  lower 
edges,  from  left  edge  with  9-inch 
ruler;  make  marks  and  draw  line 
a-h.  Eepeat  with  1-inch  ruler,  and 
draw  line  c-d.  Measure  on  left 
and  right  edges  from  upper  edge 
with  5-inch  ruler ;  make  marks  and 
draw  line  e-f.  Eepeat  with  1-inch 
ruler  and  draw  line  g-h.  Make 
cuts  indicated  at  e,  f,  g,  h.  Fold 
and  paste  into  box  shape. 

Take  other  sheet,  10"x6". 
(Fig.  2.) 

Measure  on  upper  and  lower 
edges  from  left  edge  with  4-inch  ruler;  make  marks  and 
draw  line  a-b.    Measure  on  upper  and  lower  edges  again  with 

4-inch  ruler,  but  measure  from 
points  a  and  h  instead  of  left  edge ; 
make  marks  and  draw  line  c-d. 
Cut  on  dotted  lines  a-h  and  c-d. 
Compare  rectangles,  calling  them 
rugs.  Compare  sides  of  rectangles 
by  asking  questions  about  length 
and  price  of  fringe. 
Fold  so  that  each  rectangle  is  folded  into  4  equal  parts; 
folding  long  edges  so  that  they  meet  in  each  instance.  The 
two  larger  rectangles  e 
and  /  (they  look  like  Fig. 
3  now)  are  to  be  fashion- 
ed into  rockers,  after  they 
are  pasted  in  place  under 
the  box-shape.  /7^\i        s^t 


A 

c 

E 

1 

F 

1 

'^ 

8  D 

ft9  i 


T/f  -3 


MEASURING:    SINGLE  UNIT  EULEES 


121 


'?  ¥■ 


Cut  off  the  protruding  flaps  indicated  at  1,  2,  3  and  4. 
Slope  bottom  edge  of  vertical  surface  to  look  like  rockers. 
^  Cut  g  into  strips  and  use  them  as  braces  to 

keep  rockers  in  place. 

Next  day  make  another  box  shape  out  of 
the  13-inch  square  thus : 

On    upper    and    lower   edges,   measuring 

from  left  edge  with  4-inch  ruler,  make  marks 

and  draw  line  a-h  (Fig.  4),  using  same  ruler 

on  same  edges,  but  measuring  from  right  edges  draw  line  c-d. 

Same  ruler  on  left  and  right  edges  measuring  from  upper 

edge,  draw  line  e-f. 

Same  ruler,  same  edges,  measuring  from  lower  edge,  draw 
line  g-h. 

Make  cuts  indicated  at  e,  f,  g,  h.  Fold  and  paste  into 
box  shape.  Compare  this  volume 
to  the  one  made  yesterday.  Rela- 
tion 2  to  1.  If  one  holds  a  quart 
of  beans  the  other  holds  ^  of  a 
quart. 

If  one  quart  costs  10c,  |  quart 
costs  ^  of  10c,  or  5c. 

If  one  quart  costs  12c,  ^  quart  costs  |  of  12c,  or  6c,  etc. 
Set    large    volume    on    one    end    inside    the    other    one. 
(Fig.  5.) 

LESSON  XXXIII 

Bureau— Rulers  :  3"6"9"12"— 1st  Day 

Material:    1st  day,  12-incli  square  of  stiff  paper. 

Divide  12-inch  square  into  16  small  squares,  using  the 
3''6"9'''  rulers.  Cut,  fold  and  paste  into  box  shape,  having 
dimensions  6"x6"x3". 


Viff 


122 


lATHEMATICAL  CONSTRUCTION 


Material :     2nd  day,  stiff  paper  10"x7". 
Eulers:     8"2"5''. 
Number    relations    seen :     8     5 

2     2 


rzzi 


p«5  / 


10     7 
Lay  8-inch  ruler  on  upper 
and    lower    edges,    measuring 
from   left   edge;   make   marks 
and  draw  line  a-h.     (Fig.  1.)  ' 

Lay  2-inch  ruler  in   same 
way ;  draw  line  c-d.  c 

Lay   2-inch   ruler   on    left 
and    right    edges,    measuring 
from  upper  edge;  make  marks 
and  draw  line  e-f.     Lay  2-inch  ruler  on  same  edges,  but  meas- 
ure from  lower  edge,  and  draw  line  g-h. 

Make  cuts  indicated  at  a,  b,  c,  d.  Fold  into  box  shape; 
dimensions  6''x3''x2''.  Make  two  more 
boxes  with  dimensions  6"x3"x2",  Com- 
pare their  volumes  to  each  other  and  to  the 
large  one  made  the  first  day.  Eelation  1  to 
3.  The  small  box  shapes  are  drawers  to  be 
slid  into  large  frame  G^'xCxS".  If  one 
drawer  holds  3  tablecloths,  how  many  will 
the  whole  bureau  hold?  (Ans.  3'3  table- 
cloths, or  9  tablecloths.)  If  the  tablecloths 
in  one  drawer  cost  $4.00,  how  much  does  the  bureau  full 
cost?     (Ans.  3'$4.00,  or  $12.00.) 

If  they  cost  $2.00?     (Ans.  3'$2.00,  or  $6.00.) 

Slide  the  drawers  into  the  frame.     Give  the  children  an- 


sf. 


MEASURING:    SINGLE  UNIT  RULERS 


123 


other  small  piece  of  the  paper.  From  this  let  them  make  a 
back  for  the  top  free-hand;  also  the  handles  for  the  drawers. 
Silver  paper  is  a  very  good  play  substitute  for  glass,  if  the 
children  care  to  have  a  mirror  at  the  top.  Put  on  extra  legs 
at  bottom  if  desired,  making  them  like  tiny  square  prisms. 


Material : 

&•  10" 


xro: 


LESSON  XXXIV 

A  Chiffoxier 

EuLERS :  10''2''8'' ;  2"^'%". 

1  sheet  stiff  paper  12"xl0". 

p  4  sheets  stiff  paper    8"x  6". 

1  sheet  stiff  paper    6"x  6". 

1  sheet  stiff  paper  10"x  8". 

Make  line  a-b  with  10-inch  ruler, 
measuring  from  top  of  paper,  which 
is  12"xl0''.     (Fig..l.) 

Make  line  c-d  with  2-inch  ruler, 
measuring  from  top. 

Make  line  c-f  with  8-incli  ruler, 
measuring  from  left. 

]\Iake  line  g-h  with  2-inch  ruler, 
measuring  from  left. 
Make  cuts  indicated  at  e,  f,  g,  h.     Fold  and  paste  into 
box  shape.    Large  frame  is  made.     (Fig.  1.) 

Make  line  a-b  (Fig.  2,  paper 
8"x6")  with  6-inch  ruler,  measuring 
from  left.  Use  2-inch  ruler  same 
way  to  make  line  c-d. 

Use  4-inch  ruler  to  make  line  c-f, 
measuring  from  top. 

Use    2-inch    ruler   same    way    to 


t" [ 


f/^./ 


e: 


/^/y.J. 


make  line  g-h.    Make  cuts  indicated  at  e,  f,  g,  h. 


124 


MATHEMATICAL  CONSTEUCTION 


Fold  and  paste  into  box  shape.  Make  four  of  these. 
They  are  the  four  drawers  to  the  right.  From  6-inch  square 
make  a  2-inch  cube  with  one  side  open,  using  2-  and  4-inch 
rulers.  This  is  the  little  drawer  for  the  upper  left-hand 
corner. 

Use  8-inch  ruler  to  get  line  a-b,  measuring 
from  top.  (Fig.  3.)  Use  2-inch  ruler  same 
way  to  get  line  c-d. 

Use  6 -inch  ruler,  measuring  from  left,  to 
get  line  e-f. 

Use  4-inch  ruler  same  way  to  get  line  g-h. 
Use  2-incli  ruler  same  way  to  get  line  i-j. 
Cut  out  upper  and  lower  left-hand  corners. 
Make  cuts  indicated  at  e,  f,  g,  h. 

Fold  and  paste  into  square  prism  shape.  The  side  which 
has  three  free  edges  is  the  door  of  the  cupboard.     (Fig.  4.) 

Make  legs  and  mirror  back  as  in  pre- 
vious lesson  if  desired. 

Compare  volume  of  a  and  b  (equal). 
Compare  volume  of  e  and  a  (1  to  2). 
Compare  volume  of  e  and  /  (1  to  3). 
Compare  volume  of  e  to  sum  of  a  and  b 
(1  to  4). 

Compare  volume  of  e  to  sum  of  /  and  d 
(1  to  5). 

Compare  volume  of  /  to  whole  frame  before  any  drawers 
were  put  into  it  (1  to  4). 

Compare  volume  of  a  to  volume  of  large  frame  (1  to  6). 

If  e  holds  two  of  baby's  dresses,  a  holds  2  times  2  dresses, 

or  4  dresses;  /  holds  3  times  2  dresses,  or  6  dresses;  the  sum 

of  a  and  b  holds  4  times  2  dresses,  or  8  dresses;  the  sum  of 


MEASUEING:    SINGLE  UNIT  RULERS 


125 


a,  h,  and  c,  holds  6  times  2  dresses,  or  12  dresses;  the  sum  of 
a  and  /  holds  5  times  2  dresses,  or  10  dresses,  etc. 

In  this  lesson  the  children  handled  the  rulers  2",4",G",8", 
10",  and  one  of  the  papers  was  12"  long.  They  have  dealt 
with  this  length  so  often  that  they  know  it,  even  though  the 
12-inch  ruler  is  not  in  their  hands. 

They  saw  tliat : 

'2''2"-  4"         6'2"=12" 

^/o'/-  8"         2'G''=12" 
5'2"=10'' 
They  also  saw  the  combination,  separation  and  division 
of  these  number  facts. 

LESSON  XXXV 
A  Large  Envelope 
(To  hold  weaving  strips,  or  braids  of  cord  or  raffia  until 
they  are  needed.) 

Rulers:     ir'l"8"4". 

Material:     2  sheets  manila  paper  12"x9". 
New  numerical  values  seen :     11     8 

1     1 

12     9 
c  A 


126 


MATHEMATICAL  CONSTEUCTION 


Make  line  a-b,  measuring  from  the  left  with  the  11-inch 
ruler.  Make  line  c-d  in  same  manner,  using  1-inch  ruler  to 
make  marks  c-d  and  long  ruler  to  draw  line. 

Using  8-inch  ruler,  measuring  from  top,  make  line  e-f. 

Using  -i-incli  ruler,  measuring  from  top,  make  line  g-h. 

Cut  out  corners.  Compare  them  to  each  other,  giving 
problems.     Fold  on  dotted  lines. 

Next  day  use  the  other  sheet  of  manila  paper  and  rulers 
2''4"5'"10'".    Numerical  values  seen: 
2x5"=10"         10" 
5x2"=10"  2" 

12"  (paper  is  12"  long) 


''  (paper  is  9' 
wide). 


Measuring  from  left  with 
10-ineh  ruler,  make  line  a-b. 
(Fig.  2.)  Cut  on  line.  Com- 
pare the  two  surfaces,  giving 
problems  on  varnishing  table 
tops.     (Relation  1  to  5.) 

Measuring  with  4-inch  ruler 
from  top,  make  marks  and  draw 
line  c-d. 

Paste  surface  e  on  folded 
over  flaps  of  Fig.  1,  leaving  x  for  cover  flap. 


MEASUEIXG:    SINGLE  UNIT  EULERS 


127 


LESSON  XXXVI 

Book  fok  Cuttings:  Unfolded  Sheets 

Eulers :     1"3''6"9". 

]\Iateiials :  10  sheets  grey  paper,  G"x9" ;  1  sheet  grey 
or  colored  paper,  9"xl2";  1  sheet  manila  or  colored  paper, 
G"xy".     Fifteen  inches  of  cord,  ratlia  or  ribbon  for  lacing. 

Numerical  relations  seen : 

3'1"=3''         2'3"=:6" 

6'r'=6"         3'3"=9'' 

9'r'=9" 

On  each  of  the  10  sheets  6"x9"  to  be  used  as  leaves, 
make  dots  A  and  B  measuring  with  1-inch  ruler  from  the 
upper    and    lower    left-hand    corners 

(Fig.  1). 

Make  dot  C,  using  3-inch  ruler. 
Punch  holes  where  the  dots  are. 
To   make  the  cover,  use  the  grey 
or  colored  paper,  12"x9"  (Fig.  2). 

With      the      6-inch      ruler      make 
marks  .1,  B.     Draw  line  and  cut.     Compare  the  two  surfaces, 
giving   problems   on    amount   and   cost   of 
paper,    linen   or    leatlier    for    book    covers. 
(Relations  1  to  2). 

Measuring  with  1-inch  ruler  make  line 
C-D.     Measuring  with   3-inch   ruler  make 
line  E-F. 
Fold  left  edge  to  meet  line  C-D. 
Cut  on  fold.    Xow  fold  on  line  E-F. 


3  ^ 


128 


MATHEMATICAL  CONSTEUCTION 


Using  1-inch  and  3-inch  rulers,  as  for  pages,  make  dots 
A,  B,  C,  and  punch  holes.  To  make  decorations  for  cover, 
use  manila  paper,  6"x9".     (Fig.  3.) 

Measuring  with  3-inch  ruler  from 
left  '?d^e,  make  marks  and  draw  line 
D-E. 

Cut  on  line.    Compare  the  two  sur- 
faces.   Using  1-inch  and  3-inch  rulers, 
make  dots  and  punch  holes  as  before. 
This  piece  of  paper  slips  under  the  flap  of  the  cover. 

Use  the  6-inch  square  which  is  left  folr  qny  additional 
decoration  desired.     It  can  be  cut  into  four  equal  oblongs. 
Bisect  one  of  them  and  cut  the  pieces  diagonally;  use  them 
to    strengthen    corners.      Or,    cut    the 
pieces  so  that  by  making  a  cut  \  way 
down  on  the  long  edge  and  |  way  across 
on  short  edge,  a  corner  is  removed.    Or, 
the  pieces  can  again  be  bisected,  making 
little  squares,  which  can  be  cut  in  many 
ways.    Another  pretty  decoration  is  the 

word  "Cuttings''  cut  free-hand  by  the  child,  then,  pasted  on 
the  cover. 

Lace  the  book  in  any  desired  way. 


LESSON  XXXVII 
Book  With  Folded  Leaves  for  Words 


Eulers:     Q>"\"r'. 

Material:     Six   7-inch   squares  of  white,  smooth  paper; 
one  8-inch  square  of  cover  paper. 


MEASURING 


129 


Sew  together  with  cord  12"  long. 
6 
1 


New  combinatioD  seen : 


On  each  of  the  six  squares  of  white  paper  draw  the  line 
a-h,  using  the  1-inch  ruler.     Cut  on  the  line.     These  narrow 
strips  can  be  utilized  later  in  a  weaving 
lesson.     Compare  them  with  the  large  sur- 
face.   Xow  fold  each  of  these  larije  surfaces 
so  that  their  long  edges  touch,   or  use   a 
3-inch  ruler  to  get  the  dividing  line  c-d. 
Measuring  on   this  line  from  top  and 
Q          p  bottom,  with  2-inch  ruler  make  dots  e,  f. 

Half-way  between  these  dots  (eye  judgment  only)-  make 
another  one,  dot  g.  These  are  necessary  only  on  the  page 
which  will  be  the  center  of  the  book,  so  the  child  can  see 
where  to  place  the  needle. 

Fold  the  8-inch  cover  paper  into  two  equal  oblongs. 
"With  the  2-inch  ruler  make  dots  as  at  c  and  /'.     Place 
center  dot  also.     Sew  and  decorate  cover  in  any  desired  way 


rr — T- 
:       ;  F 


This  book  is  DUE  on  the  last  date  stamped  below 


^ 


^*6^ 


UCLA-Young   Research    Library 

LB1541    .L13m 


L  009   552   992   1 


